hypothesis and conclusion conditional statements

Conditional Statement – Definition, Truth Table, Examples, FAQs

What is a conditional statement, how to write a conditional statement, what is a biconditional statement, solved examples on conditional statements, practice problems on conditional statements, frequently asked questions about conditional statements.

A conditional statement is a statement that is written in the “If p, then q” format. Here, the statement p is called the hypothesis and q is called the conclusion. It is a fundamental concept in logic and mathematics. 

Conditional statement symbol :  p → q

A conditional statement consists of two parts.

• The “if” clause, which presents a condition or hypothesis.
• The “then” clause, which indicates the consequence or result that follows if the condition is true. 

Example : If you brush your teeth, then you won’t get cavities.

Hypothesis (Condition): If you brush your teeth

Conclusion (Consequence): then you won’t get cavities 

Conditional Statement: Definition

A conditional statement is characterized by the presence of “if” as an antecedent and “then” as a consequent. A conditional statement, also known as an “if-then” statement consists of two parts:

• The “if” clause (hypothesis): This part presents a condition, situation, or assertion. It is the initial condition that is being considered.
• The “then” clause (conclusion): This part indicates the consequence, result, or action that will occur if the condition presented in the “if” clause is true or satisfied. 

Related Worksheets

Representation of Conditional Statement

The conditional statement of the form ‘If p, then q” is represented as p → q. 

It is pronounced as “p implies q.”

Different ways to express a conditional statement are:

• p implies q
• p is sufficient for q
• q is necessary for p

Parts of a Conditional Statement

There are two parts of conditional statements, hypothesis and conclusion. The hypothesis or condition will begin with the “if” part, and the conclusion or action will begin with the “then” part. A conditional statement is also called “implication.”

Conditional Statements Examples:

Example 1: If it is Sunday, then you can go to play. 

Hypothesis: If it is Sunday

Conclusion: then you can go to play. 

Example 2: If you eat all vegetables, then you can have the dessert.

Condition: If you eat all vegetables

Conclusion: then you can have the dessert 

To form a conditional statement, follow these concise steps:

Step 1 : Identify the condition (antecedent or “if” part) and the consequence (consequent or “then” part) of the statement.

Step 2 : Use the “if… then…” structure to connect the condition and consequence.

Step 3 : Ensure the statement expresses a logical relationship where the condition leads to the consequence.

Example 1 : “If you study (condition), then you will pass the exam (consequence).” 

This conditional statement asserts that studying leads to passing the exam. If you study (condition is true), then you will pass the exam (consequence is also true).

Example 2 : If you arrange the numbers from smallest to largest, then you will have an ascending order.

Hypothesis: If you arrange the numbers from smallest to largest

Conclusion: then you will have an ascending order

Truth Table for Conditional Statement

The truth table for a conditional statement is a table used in logic to explore the relationship between the truth values of two statements. It lists all possible combinations of truth values for “p” and “q” and determines whether the conditional statement is true or false for each combination. 

The truth value of p → q is false only when p is true and q is False. 

If the condition is false, the consequence doesn’t affect the truth of the conditional; it’s always true.

In all the other cases, it is true.

The truth table is helpful in the analysis of possible combinations of truth values for hypothesis or condition and conclusion or action. It is useful to understand the presence of truth or false statements. 

Converse, Inverse, and Contrapositive

The converse, inverse, and contrapositive are three related conditional statements that are derived from an original conditional statement “p → q.” 

Consider a conditional statement: If I run, then I feel great.

• Converse: 

The converse of “p → q” is “q → p.” It reverses the order of the original statement. While the original statement says “if p, then q,” the converse says “if q, then p.” 

Converse: If I feel great, then I run.

• Inverse: 

The inverse of “p → q” is “~p → ~q,” where “” denotes negation (opposite). It negates both the antecedent (p) and the consequent (q). So, if the original statement says “if p, then q,” the inverse says “if not p, then not q.”

Inverse : If I don’t run, then I don’t feel great.

• Contrapositive: 

The contrapositive of “p → q” is “~q → ~p.” It reverses the order and also negates both the statements. So, if the original statement says “if p, then q,” the contrapositive says “if not q, then not p.”

Contrapositive: If I don’t feel great, then I don’t run.

A biconditional statement is a type of compound statement in logic that expresses a bidirectional or two-way relationship between two statements. It asserts that “p” is true if and only if “q” is true, and vice versa. In symbolic notation, a biconditional statement is represented as “p ⟺ q.”

In simpler terms, a biconditional statement means that the truth of “p” and “q” are interdependent. 

If “p” is true, then “q” must also be true, and if “q” is true, then “p” must be true. Conversely, if “p” is false, then “q” must be false, and if “q” is false, then “p” must be false. 

Biconditional statements are often used to express equality, equivalence, or conditions where two statements are mutually dependent for their truth values. 

Examples : 

• I will stop my bike if and only if the traffic light is red.  
• I will stay if and only if you play my favorite song.

Facts about Conditional Statements

• The negation of a conditional statement “p → q” is expressed as “p and not q.” It is denoted as “𝑝 ∧ ∼𝑞.” 
• The conditional statement is not logically equivalent to its converse and inverse.
• The conditional statement is logically equivalent to its contrapositive. 
• Thus, we can write p → q ∼q → ∼p

In this article, we learned about the fundamentals of conditional statements in mathematical logic, including their structure, parts, truth tables, conditional logic examples, and various related concepts. Understanding conditional statements is key to logical reasoning and problem-solving. Now, let’s solve a few examples and practice MCQs for better comprehension.

Example 1: Identify the hypothesis and conclusion. 

If you sing, then I will dance.

Solution : 

Given statement: If you sing, then I will dance.

Here, the antecedent or the hypothesis is “if you sing.”

The conclusion is “then I will dance.”

Example 2: State the converse of the statement: “If the switch is off, then the machine won’t work.” 

Here, p: The switch is off

q: The machine won’t work.

The conditional statement can be denoted as p → q.

Converse of p → q is written by reversing the order of p and q in the original statement.

Converse of  p → q is q → p.

Converse of  p → q: q → p: If the machine won’t work, then the switch is off.

Example 3: What is the truth value of the given conditional statement? 

If 2+2=5 , then pigs can fly.

Solution:  

q: Pigs can fly.

The statement p is false. Now regardless of the truth value of statement q, the overall statement will be true. 

F → F = T

Hence, the truth value of the statement is true. 

Conditional Statement - Definition, Truth Table, Examples, FAQs

Attend this quiz & Test your knowledge.

What is the antecedent in the given conditional statement? If it’s sunny, then I’ll go to the beach.

A conditional statement can be expressed as, what is the converse of “a → b”, when the antecedent is true and the consequent is false, the conditional statement is.

What is the meaning of conditional statements?

Conditional statements, also known as “if-then” statements, express a cause-and-effect or logical relationship between two propositions.

When does the truth value of a conditional statement is F?

A conditional statement is considered false when the antecedent is true and the consequent is false.

What is the contrapositive of a conditional statement?

The contrapositive reverses the order of the statements and also negates both the statements. It is equivalent in truth value to the original statement.

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How to Understand ‘If-Then’ Conditional Statements: A Comprehensive Guide

In math, and even in everyday life, we often say 'if this, then that.' This is the essence of conditional statements. They set up a condition and then describe what happens if that condition is met. For instance, 'If it rains, then the ground gets wet.' These statements are foundational in math, helping us build logical arguments and solve problems. In this guide, we'll dive into the clear-cut world of conditional statements, breaking them down in both simple terms and their mathematical significance.

Step-by-step Guide: Conditional Statements

Defining Conditional Statements: A conditional statement is a logical statement that has two parts: a hypothesis (the ‘if’ part) and a conclusion (the ‘then’ part). Written symbolically, it takes the form: \( \text{If } p, \text{ then } q \) Where \( p \) is the hypothesis and \( q \) is the conclusion.

Truth Values: A conditional statement is either true or false. The only time a conditional statement is false is when the hypothesis is true, but the conclusion is false.

Converse, Inverse, and Contrapositive: 1. Converse: The converse of a conditional statement switches the hypothesis and the conclusion. For the statement “If \( p \), then \( q \)”, the converse is “If \( q \), then \( p \)”.

2. Inverse: The inverse of a conditional statement negates both the hypothesis and the conclusion. For the statement “If \( p \), then \( q \)”, the inverse is “If not \( p \), then not \( q \)”.

3. Contrapositive: The contrapositive of a conditional statement switches and negates both the hypothesis and the conclusion. For the statement “If \( p \), then \( q \)”, the contrapositive is “If not \( q \), then not \( p \)”.

Example 1: Simple Conditional Statement: “If it is raining, then the ground is wet.”

Solution: Hypothesis \(( p )\): It is raining. Conclusion \(( q )\): The ground is wet.

Example 2: Determining Truth Value Statement: “If a shape has four sides, then it is a rectangle.”

Solution: This statement is false because a shape with four sides could be a square, trapezoid, or other quadrilateral, not necessarily a rectangle.

Example 3: Converse, Inverse, and Contrapositive Statement: “If a number is even, then it is divisible by \(2\).”

Solution: Converse: If a number is divisible by \(2\), then it is even. Inverse: If a number is not even, then it is not divisible by \(2\). Contrapositive: If a number is not divisible by \(2\), then it is not even.

Practice Questions:

• Write the converse, inverse, and contrapositive for the statement: “If a bird is a penguin, then it cannot fly.”
• Determine the truth value of the statement: “If a shape has three sides, then it is a triangle.”
• For the statement “If an animal is a cat, then it is a mammal,” which of the following is its converse? a) If an animal is a mammal, then it is a cat. b) If an animal is not a cat, then it is not a mammal. c) If an animal is not a mammal, then it is not a cat.
• Converse: If a bird cannot fly, then it is a penguin. Inverse: If a bird is not a penguin, then it can fly. Contrapositive: If a bird can fly, then it is not a penguin.
• The statement is true. A shape with three sides is defined as a triangle.
• a) If an animal is a mammal, then it is a cat.

by: Effortless Math Team about 6 months ago (category: Articles )

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Conditional Statement

A conditional statement is a part of mathematical reasoning which is a critical skill that enables students to analyze a given hypothesis without any reference to a particular context or meaning. In layman words, when a scientific inquiry or statement is examined, the reasoning is not based on an individual's opinion. Derivations and proofs need a factual and scientific basis. 

Mathematical critical thinking and logical reasoning are important skills that are required to solve maths reasoning questions.

In this mini-lesson, we will explore the world of conditional statements. We will walk through the answers to the questions like what is meant by a conditional statement, what are the parts of a conditional statement, and how to create conditional statements along with solved examples and interactive questions.

Lesson Plan  

What is meant by a conditional statement.

A statement that is of the form "If p, then q" is a conditional statement. Here 'p' refers to 'hypothesis' and 'q' refers to 'conclusion'.

For example, "If Cliff is thirsty, then she drinks water."

This is a conditional statement. It is also called an implication.

'\(\rightarrow\)' is the symbol used to represent the relation between two statements. For example, A\(\rightarrow\)B. It is known as the logical connector. It can be read as A implies B. 

Here are two more conditional statement examples

Example 1: If a number is divisible by 4, then it is divisible by 2.

Example 2: If today is Monday, then yesterday was Sunday.

What Are the Parts of a Conditional Statement?

Hypothesis (if) and Conclusion (then) are the two main parts that form a conditional statement.

Let us consider the above-stated example to understand the parts of a conditional statement.

Conditional Statement : If today is Monday, then yesterday was Sunday.

Hypothesis : "If today is Monday."

Conclusion : "Then yesterday was Sunday."

On interchanging the form of statement the relationship gets changed.

To check whether the statement is true or false here, we have subsequent parts of a conditional statement. They are:

• Contrapositive

Biconditional Statement

Let us consider hypothesis as statement A and Conclusion as statement B.

Following are the observations made:

Converse of Statement

When hypothesis and conclusion are switched or interchanged, it is termed as converse statement . For example,

Conditional Statement : “If today is Monday, then yesterday was Sunday.”

Hypothesis : “If today is Monday”

Converse : “If yesterday was Sunday, then today is Monday.”

Here the conditional statement logic is, If B, then A (B → A)

Inverse of Statement

When both the hypothesis and conclusion of the conditional statement are negative, it is termed as an inverse of the statement. For example,

Conditional Statement: “If today is Monday, then yesterday was Sunday”.

Inverse : “If today is not Monday, then yesterday was not Sunday.”

Here the conditional statement logic is, If not A, then not B (~A → ~B)

Contrapositive Statement

When the hypothesis and conclusion are negative and simultaneously interchanged, then the statement is contrapositive. For example,

Contrapositive: “If yesterday was not Sunday, then today is not Monday”

Here the conditional statement logic is, if not B, then not A (~B → ~A)

The statement is a biconditional statement when a statement satisfies both the conditions as true, being conditional and converse at the same time. For example,

Biconditional : “Today is Monday if and only if yesterday was Sunday.”

Here the conditional statement logic is, A if and only if B (A ↔ B)

How to Create Conditional Statements?

Here, the point to be kept in mind is that the 'If' and 'then' part must be true.

If a number is a perfect square , then it is even.

• 'If' part is a number that is a perfect square.

Think of 4 which is a perfect square.

This has become true.

• The 'then' part is that the number should be even. 4 is even.

This has also become true. 

Thus, we have set up a conditional statement.

Let us hypothetically consider two statements, statement A and statement B. Observe the truth table for the statements:

According to the table, only if the hypothesis (A) is true and the conclusion (B) is false then, A → B will be false, or else A → B will be true for all other conditions.

• A sentence needs to be either true or false, but not both, to be considered as a mathematically accepted statement.
• Any sentence which is either imperative or interrogative or exclamatory cannot be considered a mathematically validated statement. 
• A sentence containing one or many variables is termed as an open statement. An open statement can become a statement if the variables present in the sentence are replaced by definite values.

Solved Examples

Let us have a look at a few solved examples on conditional statements.

Identify the types of conditional statements.

There are four types of conditional statements:

• If condition
• If-else condition
• Nested if-else
• If-else ladder.

Ray tells "If the perimeter of a rectangle is 14, then its area is 10."

Which of the following could be the counterexamples? Justify your decision.

a) A rectangle with sides measuring 2 and 5

b) A rectangle with sides measuring 10 and 1

c) A rectangle with sides measuring 1 and 5

d) A rectangle with sides measuring 4 and 3

a) Rectangle with sides 2 and 5: Perimeter = 14 and area = 10

Both 'if' and 'then' are true.

b) Rectangle with sides 10 and 1: Perimeter = 22 and area = 10

'If' is false and 'then' is true.

c) Rectangle with sides 1 and 5: Perimeter = 12 and area = 5

Both 'if' and 'then' are false.

d) Rectangle with sides 4 and 3: Perimeter = 14 and area = 12

'If' is true and 'then' is false.

Joe examined the set of numbers {16, 27, 24} to check if they are the multiples of 3. He claimed that they are divisible by 9. Do you agree or disagree? Justify your answer.

Conditional statement : If a number is a multiple of 3, then it is divisible by 9.

Let us find whether the conditions are true or false.

a) 16 is not a multiple of 3. Thus, the condition is false. 

16 is not divisible by 9. Thus, the conclusion is false. 

b) 27 is a multiple of 3. Thus, the condition is true.

27 is divisible by 9. Thus, the conclusion is true. 

c) 24 is a multiple of 3. Thus the condition is true.

24 is not divisible by 9. Thus the conclusion is false.

Write the converse, inverse, and contrapositive statement for the following conditional statement. 

If you study well, then you will pass the exam.

The given statement is - If you study well, then you will pass the exam.

It is of the form, "If p, then q"

The converse statement is, "You will pass the exam if you study well" (if q, then p).

The inverse statement is, "If you do not study well then you will not pass the exam" (if not p, then not q).

The contrapositive statement is, "If you did not pass the exam, then you did not study well" (if not q, then not p).

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

Let's Summarize

The mini-lesson targeted the fascinating concept of the conditional statement. The math journey around conditional statements started with what a student already knew and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever.

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

FAQs on Conditional Statement

1. what is the most common conditional statement.

'If and then' is the most commonly used conditional statement.

2. When do you use a conditional statement?

Conditional statements are used to justify the given condition or two statements as true or false.

3. What is if and if-else statement?

If is used when a specified condition is true. If-else is used when a particular specified condition is not satisfying and is false.

4. What is the symbol for a conditional statement?

'\(\rightarrow\)' is the symbol used to represent the relation between two statements. For example, A\(\rightarrow\)B. It is known as the logical connector. It can be read as A implies B.

5. What is the Contrapositive of a conditional statement?

If not B, then not A (~B → ~A)

6. What is a universal conditional statement?

Conditional statements are those statements where a hypothesis is followed by a conclusion. It is also known as an " If-then" statement. If the hypothesis is true and the conclusion is false, then the conditional statement is false. Likewise, if the hypothesis is false the whole statement is false. Conditional statements are also termed as implications.

Conditional Statement: If today is Monday, then yesterday was Sunday

Hypothesis: "If today is Monday."

Conclusion: "Then yesterday was Sunday."

If A, then B (A → B)

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• Conditional Statement

What Is A Conditional Statement?

In mathematics, we define statement as a declarative statement which may either be true or may be false. Often sentences that are mathematical in nature may not be a statement because we might not know what the variable represents. For example, 2x + 2 = 5. Now here we do not know what x represents thus if we substitute the value of x (let us consider that x = 3) i.e., 2 × 3 = 6. Therefore, it is a false statement. So, what is a conditional statement? In simple words, when through a statement we put a condition on something in return of something, we call it a conditional statement. For example, Mohan tells his friend that “if you do my homework, then I will pay you 50 dollars”. So what is happening here? Mohan is paying his friend 50 dollars but places a condition that if only he’s work will be completed by his friend. A conditional statement is made up of two parts. First, there is a hypothesis that is placed after “if” and before the comma and second is a conclusion that is placed after “then”. Here, the hypothesis will be “you do my homework” and the conclusion will be “I will pay you 50 dollars”. Now, this statement can either be true or may be false. We don’t know. 

A hypothesis is a part that is used after the 'if' and before the comma. This composes the first part of a conditional statement. For example, the statement, 'I help you get an A+ in math,' is a hypothesis because this phrase is coming in between the 'if' and the comma. So, now I hope you can spot the hypothesis in other examples of a conditional statement. Of course, you can. Here is a statement: 'If Miley gets a car, then Allie's dog will be trained,' the hypothesis here is, 'Miley gets a car.' For the statement, 'If Tom eats chocolate ice cream, then Luke eats double chocolate ice cream,' the hypothesis here is, 'Tom eats chocolate ice cream. Now it is time for you to try and locate the hypothesis for the statement, 'If the square is a rectangle, then the rectangle is a quadrilateral'?

A conclusion is a part that is used after “then”. This composes the second part of a conditional statement. For example, for the statement, “I help you get an A+ in math”, the conclusion will be “you will give me 50 dollars”. The next statement was “If Miley gets a car, then Allie's dog will be trained”, the conclusion here is Allie's dog will be trained. It is the same with the next statement and for every other conditional statement.   

How Do We Know If A Statement Is True or False? 

In mathematics, the best way we can know if a statement is true or false is by writing a mathematical proof. Before writing a proof, the mathematician must find if the statement is true or false that can be done with the help of exploration and then by finding the counterexample. Once the proof is discovered, the mathematician must communicate this discovery to those who speak the language of maths. 

Converse, Inverse, contrapositive, And Bi-conditional Statement

We usually use the term “converse” as a verb for talking and chatting and as a noun we use it to represent a brand of footwear. But in mathematics, we use it differently. Converse and inverse are the two terms that are a connected concept in the making of a conditional statement.

If we want to create the converse of a conditional statement, we just have to switch the hypothesis and the conclusion. To create the inverse of a conditional statement, we have to turn both the hypothesis and the conclusion to the negative. A contrapositive statement can be made if we first interchange the hypothesis and conclusion then make them both negative. In a bi-conditional statement, we use “if and only if” which means that the hypothesis is true only if the condition is true. For example, 

If you eat junk food, then you will gain weight is a conditional statement.

If you gained weight, then you ate junk food is a converse of a conditional statement.

If you do not eat junk food, then you will not gain weight is an inverse of a conditional statement.

If yesterday was not Monday, then today is not Tuesday is a contrapositive statement. 

Today is Tuesday if and only if yesterday was Monday is a bi-conventional statement.   

Image will be uploaded soon

A Conditional Statement Truth Table

In the table above, p→q will be false only if the hypothesis(p) will be true and the conclusion(q) will be false, or else p→q will be true. 

Conditional Statement Examples

Below, you can see some of the conditional statement examples.

Example 1) Given, P = I do my work; Q = I get the allowance

What does p→q represent?

Solution 1) In the sentence above, the hypothesis is “I do my work” and the conclusion is “ I get the allowance”. Therefore, the condition p→q represents the conditional statement, “If I do my work, then I get the allowance”. 

Example 2) Given, a = The sun is a ball of gas; b = 5 is a prime number. Write a→b in a sentence. 

Solution 2) The conditional statement a→b here is “if the sun is a ball of gas, then 5 is a prime number”.

FAQs on Conditional Statement

1. How many types of conditional statements are there?

There are basically 5 types of conditional statements.

If statement, if-else statement, nested if-else statement, if-else-if ladder, and switch statement are the basic types of conditional statements. If a function displays a statement or performs a function on the condition if the statement is true. If-else statement executes a block of code if the condition is true but if the condition is false, a new block of code is placed. The switch statement is a selection control mechanism that allows the value of a variable to change the control flow of a program. 

2. How are a conditional statement and a loop different from each other?

A conditional statement is sometimes used by a loop but a loop is of no use to a conditional statement. A conditional statement is basically a “yes” or a “no” i.e., if yes, then take the first path else take the second one. A loop is more like a cyclic chain starting from the start point i.e., if yes, then take path a, if no, take path b and it comes back to the start point. 

Conditional statement executes a statement based on a condition without causing any repetition. 

A loop executes a statement repeatedly. There are two loop variables i.e., for loop and while loop.

Calcworkshop

Conditional Statement If Then's Defined in Geometry - 15+ Examples!

// Last Updated: January 21, 2020 - Watch Video //

In today’s geometry lesson , you’re going to learn all about conditional statements!

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

We’re going to walk through several examples to ensure you know what you’re doing.

In addition, this lesson will prepare you for deductive reasoning and two column proofs later on.

Here we go!

What are Conditional Statements?

To better understand deductive reasoning, we must first learn about conditional statements.

A conditional statement has two parts: hypothesis ( if ) and conclusion ( then ).

In fact, conditional statements are nothing more than “If-Then” statements!

Sometimes a picture helps form our hypothesis or conclusion. Therefore, we sometimes use Venn Diagrams to visually represent our findings and aid us in creating conditional statements.

But to verify statements are correct, we take a deeper look at our if-then statements. This is why we form the converse , inverse , and contrapositive of our conditional statements.

What is the Converse of a Statement?

Well, the converse is when we switch or interchange our hypothesis and conclusion.

Conditional Statement : “If today is Wednesday, then yesterday was Tuesday.”

Hypothesis : “If today is Wednesday” so our conclusion must follow “Then yesterday was Tuesday.”

So the converse is found by rearranging the hypothesis and conclusion, as Math Planet accurately states.

Converse : “If yesterday was Tuesday, then today is Wednesday.”

What is the Inverse of a Statement?

Now the inverse of an If-Then statement is found by negating (making negative) both the hypothesis and conclusion of the conditional statement.

So using our current conditional statement, “If today is Wednesday, then yesterday was Tuesday”.

Inverse : “If today is not Wednesday, then yesterday was not Tuesday.”

What is a Contrapositive?

And the contrapositive is formed by interchanging the hypothesis and conclusion and then negating both.

Contrapositive : “If yesterday was not Tuesday, then today is not Wednesday”

What is a Biconditional Statement?

A statement written in “if and only if” form combines a reversible statement and its true converse. In other words the conditional statement and converse are both true.

Continuing with our initial condition, “If today is Wednesday, then yesterday was Tuesday.”

Biconditional : “Today is Wednesday if and only if yesterday was Tuesday.”

Examples of Conditional Statements

In the video below we will look at several harder examples of how to form a proper statement, converse, inverse, and contrapositive. And here’s a big hint…

Whenever you see “con” that means you switch! It’s like being a con-artist!

Moreover, we will detail the process for coming up with reasons for our conclusions using known postulates. We will review the ten postulates that we have learned so far, and add a few more problems dealing with perpendicular lines, planes, and perpendicular bisectors.

After this lesson, we will be ready to tackle deductive reasoning head-on, and feel confident as we march onward toward learning two-column proofs!

Conditional Statements – Lesson & Examples (Video)

• Introduction to conditional statements
• 00:00:25 – What are conditional statements, converses, and biconditional statements? (Examples #1-2)
• 00:05:21 – Understanding venn diagrams (Examples #3-4)
• 00:11:07 – Supply the missing venn diagram and conditional statement for each question (Examples #5-8)
• Exclusive Content for Member’s Only
• 00:17:48 – Write the statement and converse then determine if they are reversible (Examples #9-12)
• 00:29:17 – Understanding the inverse, contrapositive, and symbol notation
• 00:35:33 – Write the statement, converse, inverse, contrapositive, and biconditional statements for each question (Examples #13-14)
• 00:45:40 – Using geometry postulates to verify statements (Example #15)
• 00:53:23 – What are perpendicular lines, perpendicular planes and the perpendicular bisector?
• 00:56:26 – Using the figure, determine if the statement is true or false (Example #16)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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17.6: Truth Tables: Conditional, Biconditional

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• David Lippman
• Pierce College via The OpenTextBookStore

We discussed conditional statements earlier, in which we take an action based on the value of the condition. We are now going to look at another version of a conditional, sometimes called an implication, which states that the second part must logically follow from the first.

Conditional

A conditional is a logical compound statement in which a statement \(p\), called the antecedent, implies a statement \(q\), called the consequent.

A conditional is written as \(p \rightarrow q\) and is translated as "if \(p\), then \(q\)".

The English statement “If it is raining, then there are clouds is the sky” is a conditional statement. It makes sense because if the antecedent “it is raining” is true, then the consequent “there are clouds in the sky” must also be true.

Notice that the statement tells us nothing of what to expect if it is not raining; there might be clouds in the sky, or there might not. If the antecedent is false, then the consquent becomes irrelevant.

Suppose you order a team jersey online on Tuesday and want to receive it by Friday so you can wear it to Saturday’s game. The website says that if you pay for expedited shipping, you will receive the jersey by Friday. In what situation is the website telling a lie?

There are four possible outcomes:

1) You pay for expedited shipping and receive the jersey by Friday

2) You pay for expedited shipping and don’t receive the jersey by Friday

3) You don’t pay for expedited shipping and receive the jersey by Friday

4) You don’t pay for expedited shipping and don’t receive the jersey by Friday

Only one of these outcomes proves that the website was lying: the second outcome in which you pay for expedited shipping but don’t receive the jersey by Friday. The first outcome is exactly what was promised, so there’s no problem with that. The third outcome is not a lie because the website never said what would happen if you didn’t pay for expedited shipping; maybe the jersey would arrive by Friday whether you paid for expedited shipping or not. The fourth outcome is not a lie because, again, the website didn’t make any promises about when the jersey would arrive if you didn’t pay for expedited shipping.

It may seem strange that the third outcome in the previous example, in which the first part is false but the second part is true, is not a lie. Remember, though, that if the antecedent is false, we cannot make any judgment about the consequent. The website never said that paying for expedited shipping was the only way to receive the jersey by Friday.

A friend tells you “If you upload that picture to Facebook, you’ll lose your job.” Under what conditions can you say that your friend was wrong?

1) You upload the picture and lose your job

2) You upload the picture and don’t lose your job

3) You don’t upload the picture and lose your job

4) You don’t upload the picture and don’t lose your job

There is only one possible case in which you can say your friend was wrong: the second outcome in which you upload the picture but still keep your job. In the last two cases, your friend didn’t say anything about what would happen if you didn’t upload the picture, so you can’t say that their statement was wrong. Even if you didn’t upload the picture and lost your job anyway, your friend never said that you were guaranteed to keep your job if you didn’t upload the picture; you might lose your job for missing a shift or punching your boss instead.

In traditional logic, a conditional is considered true as long as there are no cases in which the antecedent is true and the consequent is false.

Truth table for the conditional

\(\begin{array}{|c|c|c|} \hline p & q & p \rightarrow q \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}\)

Again, if the antecedent \(p\) is false, we cannot prove that the statement is a lie, so the result of the third and fourth rows is true.

Construct a truth table for the statement \((m \wedge \sim p) \rightarrow r\)

We start by constructing a truth table with 8 rows to cover all possible scenarios. Next, we can focus on the antecedent, \(m \wedge \sim p\).

\(\begin{array}{|c|c|c|} \hline m & p & r \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \end{array}\)

\(\begin{array}{|c|c|c|c|} \hline m & p & r & \sim p \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}\)

\(\begin{array}{|c|c|c|c|c|} \hline m & p & r & \sim p & m \wedge \sim p \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \end{array}\)

Now we can create a column for the conditional. Because it can be confusing to keep track of all the Ts and \(\mathrm{Fs}\), why don't we copy the column for \(r\) to the right of the column for \(m \wedge \sim p\) ? This makes it a lot easier to read the conditional from left to right.

\(\begin{array}{|c|c|c|c|c|c|c|} \hline m & p & r & \sim p & m \wedge \sim p & r & (m \wedge \sim p) \rightarrow r \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}\)

When \(m\) is true, \(p\) is false, and \(r\) is false- -the fourth row of the table-then the antecedent \(m \wedge \sim p\) will be true but the consequent false, resulting in an invalid conditional; every other case gives a valid conditional.

If you want a real-life situation that could be modeled by \((m \wedge \sim p) \rightarrow r\), consider this: let \(m=\) we order meatballs, \(p=\) we order pasta, and \(r=\) Rob is happy. The statement \((m \wedge \sim p) \rightarrow r\) is "if we order meatballs and don't order pasta, then Rob is happy". If \(m\) is true (we order meatballs), \(p\) is false (we don't order pasta), and \(r\) is false (Rob is not happy), then the statement is false, because we satisfied the antecedent but Rob did not satisfy the consequent.

For any conditional, there are three related statements, the converse, the inverse, and the contrapositive.

Related Statments

The original conditional is \(\quad\) "if \(p,\) then \(q^{\prime \prime} \quad p \rightarrow q\)

The converse is \(\quad\) "if \(q,\) then \(p^{\prime \prime} \quad q \rightarrow p\)

The inverse is \(\quad\) "if not \(p,\) then not \(q^{\prime \prime} \quad \sim p \rightarrow \sim q\)

The contrapositive is "if not \(q,\) then not \(p^{\prime \prime} \quad \sim q \rightarrow \sim p\)

Consider again the conditional “If it is raining, then there are clouds in the sky.” It seems reasonable to assume that this is true.

The converse would be “If there are clouds in the sky, then it is raining.” This is not always true.

The inverse would be “If it is not raining, then there are not clouds in the sky.” Likewise, this is not always true.

The contrapositive would be “If there are not clouds in the sky, then it is not raining.” This statement is true, and is equivalent to the original conditional.

Looking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent, and that the converse and inverse are logically equivalent.

Equivalence

A conditional statement and its contrapositive are logically equivalent.

The converse and inverse of a conditional statement are logically equivalent.

In other words, the original statement and the contrapositive must agree with each other; they must both be true, or they must both be false. Similarly, the converse and the inverse must agree with each other; they must both be true, or they must both be false.

Be aware that symbolic logic cannot represent the English language perfectly. For example, we may need to change the verb tense to show that one thing occurred before another.

Suppose this statement is true: “If I eat this giant cookie, then I will feel sick.” Which of the following statements must also be true?

• If I feel sick, then I ate that giant cookie.
• If I don’t eat this giant cookie, then I won’t feel sick.
• If I don’t feel sick, then I didn’t eat that giant cookie.
• This is the converse, which is not necessarily true. I could feel sick for some other reason, such as drinking sour milk.
• This is the inverse, which is not necessarily true. Again, I could feel sick for some other reason; avoiding the cookie doesn’t guarantee that I won’t feel sick.
• This is the contrapositive, which is true, but we have to think somewhat backwards to explain it. If I ate the cookie, I would feel sick, but since I don’t feel sick, I must not have eaten the cookie.

Notice again that the original statement and the contrapositive have the same truth value (both are true), and the converse and the inverse have the same truth value (both are false).

Try it Now 5

“If you microwave salmon in the staff kitchen, then I will be mad at you.” If this statement is true, which of the following statements must also be true?

• If you don’t microwave salmon in the staff kitchen, then I won’t be mad at you.
• If I am not mad at you, then you didn’t microwave salmon in the staff kitchen.
• If I am mad at you, then you microwaved salmon in the staff kitchen.

Choice b is correct because it is the contrapositive of the original statement.

Consider the statement “If you park here, then you will get a ticket.” What set of conditions would prove this statement false?

• You don’t park here and you get a ticket.
• You don’t park here and you don’t get a ticket.
• You park here and you don’t get a ticket.

The first two statements are irrelevant because we don’t know what will happen if you park somewhere else. The third statement, however contradicts the conditional statement “If you park here, then you will get a ticket” because you parked here but didn’t get a ticket. This example demonstrates a general rule; the negation of a conditional can be written as a conjunction: “It is not the case that if you park here, then you will get a ticket” is equivalent to “You park here and you do not get a ticket.”

The Negation of a Conditional

The negation of a conditional statement is logically equivalent to a conjunction of the antecedent and the negation of the consequent.

\(\sim(p \rightarrow q)\) is equivalent to \(p \wedge \sim q\)

Which of the following statements is equivalent to the negation of “If you don’t grease the pan, then the food will stick to it” ?

• I didn’t grease the pan and the food didn’t stick to it.
• I didn’t grease the pan and the food stuck to it.
• I greased the pan and the food didn’t stick to it.
• This is correct; it is the conjunction of the antecedent and the negation of the consequent. To disprove that not greasing the pan will cause the food to stick, I have to not grease the pan and have the food not stick.
• This is essentially the original statement with no negation; the “if…then” has been replaced by “and”.
• This essentially agrees with the original statement and cannot disprove it.

Try it Now 6

“If you go swimming less than an hour after eating lunch, then you will get cramps.” Which of the following statements is equivalent to the negation of this statement?

• I went swimming more than an hour after eating lunch and I got cramps.
• I went swimming less than an hour after eating lunch and I didn’t get cramps.
• I went swimming more than an hour after eating lunch and I didn’t get cramps.

Choice b is equivalent to the negation; it keeps the first part the same and negates the second part.

In everyday life, we often have a stronger meaning in mind when we use a conditional statement. Consider “If you submit your hours today, then you will be paid next Friday.” What the payroll rep really means is “If you submit your hours today, then you will be paid next Friday, and if you don’t submit your hours today, then you won’t be paid next Friday.” The conditional statement if t , then p also includes the inverse of the statement: if not t , then not p . A more compact way to express this statement is “You will be paid next Friday if and only if you submit your timesheet today.” A statement of this form is called a biconditional .

Biconditional

A biconditional is a logical conditional statement in which the antecedent and consequent are interchangeable.

A biconditional is written as \(p \leftrightarrow q\) and is translated as " \(p\) if and only if \(q^{\prime \prime}\).

Because a biconditional statement \(p \leftrightarrow q\) is equivalent to \((p \rightarrow q) \wedge(q \rightarrow p),\) we may think of it as a conditional statement combined with its converse: if \(p\), then \(q\) and if \(q\), then \(p\). The double-headed arrow shows that the conditional statement goes from left to right and from right to left. A biconditional is considered true as long as the antecedent and the consequent have the same truth value; that is, they are either both true or both false.

Truth table for the biconditional

\(\begin{array}{|c|c|c|} \hline p & q & p \leftrightarrow q \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}\)

Notice that the fourth row, where both components are false, is true; if you don’t submit your timesheet and you don’t get paid, the person from payroll told you the truth.

Suppose this statement is true: “The garbage truck comes down my street if and only if it is Thursday morning.” Which of the following statements could be true?

• It is noon on Thursday and the garbage truck did not come down my street this morning.
• It is Monday and the garbage truck is coming down my street.
• It is Wednesday at 11:59PM and the garbage truck did not come down my street today.
• This cannot be true. This is like the second row of the truth table; it is true that I just experienced Thursday morning, but it is false that the garbage truck came.
• This cannot be true. This is like the third row of the truth table; it is false that it is Thursday, but it is true that the garbage truck came.
• This could be true. This is like the fourth row of the truth table; it is false that it is Thursday, but it is also false that the garbage truck came, so everything worked out like it should.

Try it Now 7

Suppose this statement is true: “I wear my running shoes if and only if I am exercising.” Determine whether each of the following statements must be true or false.

• I am exercising and I am not wearing my running shoes.
• I am wearing my running shoes and I am not exercising.
• I am not exercising and I am not wearing my running shoes.

Choices a & b are false; c is true.

Create a truth table for the statement \((A \vee B) \leftrightarrow \sim C\)

Whenever we have three component statements, we start by listing all the possible truth value combinations for \(A, B,\) and \(C .\) After creating those three columns, we can create a fourth column for the antecedent, \(A \vee B\). Now we will temporarily ignore the column for \(C\) and focus on \(A\) and \(B\), writing the truth values for \(A \vee B\).

\(\begin{array}{|c|c|c|} \hline A & B & C \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \end{array}\)

\(\begin{array}{|c|c|c|c|} \hline A & B & C & A \vee B \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \end{array}\)

Next we can create a column for the negation of \(C\). (Ignore the \(A \vee B\) column and simply negate the values in the \(C\) column.)

\(\begin{array}{|c|c|c|c|c|} \hline A & B & C & A \vee B & \sim C \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}\)

Finally, we find the truth values of \((A \vee B) \leftrightarrow \sim C\). Remember, a biconditional is true when the truth value of the two parts match, but it is false when the truth values do not match.

\(\begin{array}{|c|c|c|c|c|c|} \hline A & B & C & A \vee B & \sim C & (A \vee B) \leftrightarrow \sim C \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \end{array}\)

To illustrate this situation, suppose your boss needs you to do either project \(A\) or project \(B\) (or both, if you have the time). If you do one of the projects, you will not get a crummy review ( \(C\) is for crummy). So \((A \vee B) \leftrightarrow \sim C\) means "You will not get a crummy review if and only if you do project \(A\) or project \(B\)." Looking at a few of the rows of the truth table, we can see how this works out. In the first row, \(A, B,\) and \(C\) are all true: you did both projects and got a crummy review, which is not what your boss told you would happen! That is why the final result of the first row is false. In the fourth row, \(A\) is true, \(B\) is false, and \(C\) is false: you did project \(A\) and did not get a crummy review. This is what your boss said would happen, so the final result of this row is true. And in the eighth row, \(A, B\), and \(C\) are all false: you didn't do either project and did not get a crummy review. This is not what your boss said would happen, so the final result of this row is false. (Even though you may be happy that your boss didn't follow through on the threat, the truth table shows that your boss lied about what would happen.)

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Conditional Statements

Two common types of statements found in the study of logic are conditional and biconditional statements. They are formed by combining two statements which we then we call compound statements. What if we were to say, "If it snows, then we don't go outside." This is two statements combined. They are often called if-then statements. As in, "IF it snows, THEN we don't go outside." They are a fundamental building block of computer programming.

Writing conditional statements

A statement written in if-then format is a conditional statement.

It looks like

This represents the conditional statement:

"If p then q ."

A conditional statement is also called an implication.

If a closed shape has three sides, then it is a triangle.

The part of the statement that follows the "if" is called the hypothesis. The part of the statement that follows the "then" is the conclusion.

So in the above statement,

If a closed shape has three sides, (this is the hypothesis)

Then it is a triangle. (this is the conclusion)

Identify the hypothesis and conclusion of the following conditional statement.

A polygon is a hexagon if it has six sides.

Hypothesis: The polygon has six sides.

Conclusion: It is a hexagon.

The hypothesis does not always come first in a conditional statement. You must read it carefully to determine which part of the statement is the hypothesis and which part is the conclusion.

Truth table for conditional statement

The truth table for any two given inputs, say A and B , is given by:

• If A and B are both true, then A → B is true.
• If A is true and B is false, then A → B is false.
• If A is false and B is true, then A → B is true.
• If A and B are both false, then A → B is true.

Take our conditional statement that if it snows, we do not go outside.

If it is snowing ( A is true) and we do go outside ( B is false), then the statement A → B is false.

If it is not snowing ( A is false), it doesn't matter if we go outside or not ( B is true or false), because A → B is impossible to determine if A is false, so the statement A → B can still be true.

Biconditional statements

A biconditional statement is a combination of a statement and its opposite written in the format of "if and only if."

For example, "Two line segments are congruent if and only if they are the same length."

This is a combination of two conditional statements.

"Two line segments are congruent if they are the same length."

"Two line segments are the same length if they are congruent."

A biconditional statement is true if and only if both the conditional statements are true.

Biconditional statements are represented by the symbol:

p ↔ q

p ↔ q = p → q ∧ q → p

Writing biconditional statements

Write the two conditional statements that make up this biconditional statement:

I am punctual if and only if I am on time to school every day.

The two conditional statements that have to be true to make this statement true are:

• I am punctual if I am on time to school every day.
• I am on time to school every day if I am punctual.

A rectangle is a square if and only if the adjacent sides are congruent.

• If the adjacent sides of a rectangle are congruent then it is a square.
• If a rectangle is a square then the adjacent sides are congruent.

Topics related to the Conditional Statements

Conjunction

Counter Example

Biconditional Statement

Flashcards covering the Conditional Statements

Symbolic Logic Flashcards

Introduction to Proofs Flashcards

Practice tests covering the Conditional Statements

Introduction to Proofs Practice Tests

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If-then statement

• Logical correct I
• Logical correct II

When we previously discussed inductive reasoning we based our reasoning on examples and on data from earlier events. If we instead use facts, rules and definitions then it's called deductive reasoning.

We will explain this by using an example.

If you get good grades then you will get into a good college.

The part after the "if": you get good grades - is called a hypotheses and the part after the "then" - you will get into a good college - is called a conclusion.

Hypotheses followed by a conclusion is called an If-then statement or a conditional statement.

This is noted as

$$p \to q$$

This is read - if p then q.

A conditional statement is false if hypothesis is true and the conclusion is false. The example above would be false if it said "if you get good grades then you will not get into a good college".

If we re-arrange a conditional statement or change parts of it then we have what is called a related conditional.

Our conditional statement is: if a population consists of 50% men then 50% of the population must be women.

If we exchange the position of the hypothesis and the conclusion we get a converse statemen t: if a population consists of 50% women then 50% of the population must be men.

$$q\rightarrow p$$

If both statements are true or if both statements are false then the converse is true. A conditional and its converse do not mean the same thing

If we negate both the hypothesis and the conclusion we get a inverse statemen t: if a population do not consist of 50% men then the population do not consist of 50% women.

$$\sim p\rightarrow \: \sim q$$

The inverse is not true juest because the conditional is true. The inverse always has the same truth value as the converse.

We could also negate a converse statement, this is called a contrapositive statemen t:  if a population do not consist of 50% women then the population do not consist of 50% men.

$$\sim q\rightarrow \: \sim p$$

The contrapositive does always have the same truth value as the conditional. If the conditional is true then the contrapositive is true.

A pattern of reaoning is a true assumption if it always lead to a true conclusion. The most common patterns of reasoning are detachment and syllogism.

If we turn of the water in the shower, then the water will stop pouring.

If we call the first part p and the second part q then we know that p results in q. This means that if p is true then q will also be true. This is called the law of detachment and is noted:

$$\left [ (p \to q)\wedge p \right ] \to q$$

The law of syllogism tells us that if p → q and q → r then p → r is also true.

This is noted:

$$\left [ (p \to q)\wedge (q \to r ) \right ] \to (p \to r)$$

If the following statements are true:

If we turn of the water (p), then the water will stop pouring (q). If the water stops pouring (q) then we don't get wet any more (r).

Then the law of syllogism tells us that if we turn of the water (p) then we don't get wet (r) must be true.

Video lesson

Write a converse, inverse and contrapositive to the conditional

"If you eat a whole pint of ice cream, then you won't be hungry"

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• Open access
• Published: 17 April 2024

Decline of a distinct coral reef holobiont community under ocean acidification

• Jake Williams 1 , 2 ,
• Nathalie Pettorelli 2 ,
• Aaron C. Hartmann 3 ,
• Robert A. Quinn 4 ,
• Laetitia Plaisance 5 , 6 ,
• Michael O’Mahoney 6 ,
• Chris P. Meyer 6 ,
• Katharina E. Fabricius 7 ,
• Nancy Knowlton 6 &
• Emma Ransome 1  

Microbiome volume  12 , Article number:  75 ( 2024 ) Cite this article

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Microbes play vital roles across coral reefs both in the environment and inside and upon macrobes (holobionts), where they support critical functions such as nutrition and immune system modulation. These roles highlight the potential ecosystem-level importance of microbes, yet most knowledge of microbial functions on reefs is derived from a small set of holobionts such as corals and sponges. Declining seawater pH — an important global coral reef stressor — can cause ecosystem-level change on coral reefs, providing an opportunity to study the role of microbes at this scale. We use an in situ experimental approach to test the hypothesis that under such ocean acidification (OA), known shifts among macrobe trophic and functional groups may drive a general ecosystem-level response extending across macrobes and microbes, leading to reduced distinctness between the benthic holobiont community microbiome and the environmental microbiome.

We test this hypothesis using genetic and chemical data from benthic coral reef community holobionts sampled across a pH gradient from CO 2 seeps in Papua New Guinea. We find support for our hypothesis; under OA, the microbiome and metabolome of the benthic holobiont community become less compositionally distinct from the sediment microbiome and metabolome, suggesting that benthic macrobe communities are colonised by environmental microbes to a higher degree under OA conditions. We also find a simplification and homogenisation of the benthic photosynthetic community, and an increased abundance of fleshy macroalgae, consistent with previously observed reef microbialisation.

Conclusions

We demonstrate a novel structural shift in coral reefs involving macrobes and microbes: that the microbiome of the benthic holobiont community becomes less distinct from the sediment microbiome under OA. Our findings suggest that microbialisation and the disruption of macrobe trophic networks are interwoven general responses to environmental stress, pointing towards a universal, undesirable, and measurable form of ecosystem change.

Video Abstract

Increased atmospheric greenhouse gas concentrations are leading to increased partial pressure of CO 2 (pCO 2 ) and reduced pH in the surface water of the oceans, which have absorbed 25% of all anthropogenic CO 2 to date [ 1 ]. The impacts of this phenomenon, known as ocean acidification (OA) [ 2 ], are predicted to be particularly severe for coral reefs due to declines in net calcification by organisms, which impacts reef structure and diversity [ 3 ]. Severe impacts on coral reefs are concerning as these ecosystems host vast biodiversity and provide significant ecosystem services to humanity (e.g. coastal protection, food security, and new medicines), and other significant terrestrial and pelagic ocean ecosystems are also directly dependent on their functioning [ 4 ]. Being able to predict how coral ecosystems change under OA, and what this means for their ability to continue providing ecosystem services to society, is therefore a global environmental and social priority.

Microbes play important roles on coral reefs, from carrying out nutrient cycling, which contributes to ecosystem productivity in nutrient poor waters [ 5 ], to their roles in immunity and defence for a wide range of reef invertebrates, including cnidarians, sponges, molluscs, and echinoderms [ 6 , 7 , 8 , 9 ]. These interactions between macrobes and microbes are known to shift in response to environmental change, such as OA. Such shifts include changing microbial associations with particular macrobes, such as corals [ 10 , 11 ] and sponges [ 12 ], and changes in the functional profile of specific microbiomes, such as alterations in nitrogen (N 2 ) fixation by coral-associated bacteria [ 13 ] and shifting metabolic activity of free-living bacteria in the water column [ 14 ]. However, the traditional focus of research on individual macrobes and their microbiomes (holobionts) makes it challenging to scale up our understanding to ecosystem-level shifts in macrobe-microbe interactions, which, though often overlooked, can play a significant role in driving ecosystem-level change [ 11 , 15 , 16 ].

Ecosystem-level impacts of OA are expected to result from interaction-mediated changes at the community level, driven by different physiochemical effects of OA at the level of organism metabolism [ 2 , 17 ]. Well-established organism-level effects of OA include benefits to fleshy algae and other photosynthetic organisms from the resource effect of increased pCO 2 [ 2 , 18 ], which can result in enhanced net dissolved organic carbon (DOC) release [ 19 ]. In contrast, calcifiers suffer from increased costs of calcification due to reduced pH [ 2 ]. These differentiated organism-level impacts can shift ecological interactions between taxa [ 2 , 17 ], with cascading effects on energy flows through an ecosystem via altered nutrition and metabolism (i.e. altered trophodynamics [ 20 , 21 ]). Such indirect cascading effects can impact entire trophic networks [ 22 ], including macrobe-microbe trophic interactions [ 23 ]. However, our understanding of the indirect effects of ocean acidification on coral reefs remains limited [ 24 ].

Here, we propose that a previously observed ecosystem-level impact of specific stressors on coral reefs — microbialisation — may be generalisable to OA. Microbialisation refers to an increase in microbial biomass resulting from a reallocation of energy from macrobes to microbes [ 23 , 25 ]. On coral reefs, the proximal causes of microbialisation have been proposed to be overfishing and eutrophication, which facilitate the enhanced growth of fleshy algae and cause an increased release of dissolved organic carbon (DOC) [ 26 ]. Elevated DOC has been proposed to increase microbial biomass and disease (the DDAM (DOC, disease, algae, microbes) positive feedback loop) [ 26 ].

We build on a commonality between proposed organism-level mechanisms of microbialisation, namely that macrobe communities are expected to be more vulnerable to stressors than microbial communities [ 26 , 27 ], an observation both supported in general [ 28 ] and in the case of OA in particular [ 29 ]. This greater vulnerability of macrobes should lead to declines in some benthic taxa, such as calcified algae, soft and hard corals, sponges [ 18 ], and calcified grazers (e.g. gastropods [ 30 ]), already documented with OA, and to the disruption of associated macrobe trophic pathways. Examples of these trophic pathways include calcified grazers consuming algal communities (which controls algal proliferation) [ 31 ], and sponges removing vast quantities of dissolved organic matter (DOM) from the water column and converting it into food for higher trophic levels (e.g. polychaetes and brittle stars) via the sponge loop [ 32 ]. With declines in benthic taxa, some of the free energy cycled through such pathways can become available to less impacted taxa, such as environmental microbes [ 20 , 33 ], and is expected to drive the ecological release of such taxa in both density and niche expansion [ 34 ].

We hypothesise that we will observe a decline in the compositional and functional distinctness between the benthic holobiont community microbiome and the environmental sediment-dwelling microbiome as a result of microbialisation occurring under OA. Almost all macrobes are holobionts with a symbiotic microbiome [ 35 , 36 ], and therefore, microbialisation has the potential to impact the microbiome of the entire holobiont community (recently referred to as the eco-holobiont [ 37 ]). The process of microbialisation should result in decreased compositional and functional distinctness between the benthic holobiont community microbiome and the sediment microbiome through two mechanisms. Firstly, direct impacts on macrobes may alter host metabolism and reduce the resources or habitat available to the holobiont community microbiome, leading to opportunistic environmental microbes displacing holobiont-specialised microbes (e.g. [ 38 ]). Secondly, increased abundances of environmental microbes and increased microbe trophic interactions with macrobes (expected due to the loss of macrobe competitors [ 39 ]) should lead to increased opportunities for colonisation of the holobiome by environmental microbes [ 40 ], including those in the sediment microbiome.

To test whether this decline in the distinctness of the benthic holobiont community microbiome occurs, we generated a unique multiomic dataset from autonomous reef monitoring structures (ARMS) deployed on a natural OA gradient caused by CO 2 seeps. ARMS are three-dimensional, artificial settlement structures designed to mimic the structural complexity of coral reef environments, which are increasingly used to monitor coral reefs across the globe (e.g. [ 41 , 42 , 43 ]). They enable the non-destructive and standardised sampling of a large proportion of reef diversity that is often not studied, including algae and cryptic benthic invertebrates such as sponges, cnidarians, bryozoans, and annelids [ 44 ], alongside their associated microbes, and sediment-dwelling environmental microbes. ARMS allow us to investigate OA using a holistic microbial ecosystem approach that integrates across scales from individual microbes and benthic holobionts, to neighbouring holobionts that, in turn, interact with and influence successively larger and more complex communities. We studied the effects of OA using an in situ experimental approach at a location with naturally occurring CO 2 seeps that produce pH and pCO 2 gradients. These seeps have been intensively studied because they can help predict future ocean conditions under OA [ 45 ].

Multiomics, in this case metabarcoding and metabolomics, provides a powerful toolkit to investigate the effects of stressors across entire communities. Metabarcoding provides data on genetic community composition and diversity (i.e. compositional metrics) [ 46 ], while metabolomics provides comparable data on biochemical composition and diversity of the metabolome (i.e. functional metrics) [ 47 ]. We first confirm that the expected photosynthetic community shifts take place (as previously documented [ 18 ]), consistent with microbialisation. We then analyse the ecosystem-level effects of OA by comparing the distinctness of the sediment microbiome to: (1) the benthic holobiont community microbiome and (2) individual sponge microbiomes. We expected OA to cause a decline in distinctness as a result of ecosystem microbialisation.

Materials and methods

Experimental design and sampling.

This study was carried out at CO 2 seeps and adjacent control sites in Milne Bay Province, Papua New Guinea (Fig.  1 ), located at 9° s latitude in the heart of the Coral Triangle. The two studied seep localities (Upa-Upasina and Dobu) are located along an active tectonic fault where > 99% CO 2 gas has been streaming though the reef substrata at ambient temperature (28.6–29.7 °C) for at least 100 years and probably much longer [ 18 ]. The reefs surrounding the seeps are under low anthropogenic pressure and have been used to study ocean acidification for the last decade (e.g. [ 18 , 48 ]). The six study sites (two localities, each with three pH levels) exhibit similar geomorphology, temperature and salinity, but contrasting pCO 2 and pH [ 18 ]. Water temperature, pH, salinity, and pressure at the study sites have been monitored regularly (2010–2016), making this an ideal location to study the isolated impacts of OA.

Location of the study localities, Dobu and Upa-Upasina, in Milne Bay Province, Papua New Guinea ( A & B )

In April, 2012, eighteen autonomous reef monitoring structures (ARMS; Fig.  2 A) were deployed at 3 m depth adjacent to coral reefs at 3 pH levels at each of the 2 localities (mean pH: control 7.99 & 8.01, medium 7.85 at both localities, and low 7.64 & 7.75; n  = 3 per pH level [ 48 ]). ARMS were collected from the seafloor (Fig.  2 B) after 31 months, in November 2014. A 106-μm nitex-lined crate was placed over each ARMS on the seafloor, and they were together returned to the surface, after which each ARMS was placed in an individual holding tank with 45-μm filtered aerated seawater. ARMS were then transported to shore where they were sampled rapidly to minimise molecular and chemical degradation; transportation time of ARMS was < 20 min, and processing of each sample type was completed within 1.5 h.

Images of Autonomous Reef Monitoring Structures (ARMS) and ARMS plates. Images show an ARMS in situ ( A ) and nestled within the reef after 2.5 years of deployment ( B ). A light-exposed ARMS top plate, from which the benthic photosynthetic community was sampled, can be seen in ( C ). A Tethya sp. sponge (1) and a Halisarca sp. sponge (2), commonly observed on recovered ARMS, can be seen in ( D ). An internal light-limited ARMS plate with crossbars, which create sheltered conditions and mimic the natural reef, can be seen in ( E )

Sample extraction and multiomics

The standard ARMS processing protocol [ 44 ] was modified to test our specific hypothesis. From each ARMS unit, five fractions were collected: the benthic photosynthetic community, the benthic holobiont community, the sediment, Halisarca sp. sponge, and Tethya sp. sponge. To do this, ARMS were removed from their holding tanks, and the 9 plates (17 plate surfaces as there is no accessible bottom surface to the bottom plate) were separated and rinsed to dislodge loosely attached organisms. The water and previously trapped sediment in the holding tank were retained.

First, the benthic photosynthetic community was sampled by randomly subsampling (4 × 1 cm 2 ) the top surface of the top ARMS plate (e.g. Figure  2 C), which resembles the algal community found on exposed rocky substrates (e.g. macroalgae, algal turf, calcified algae, and cyanobacterial mats). Second, the two sponge fractions ( Halisarca sp. and Tethya sp., e.g. Figure  2 D) were generated by sampling morphologically identified sponges from the internal plates. Both are low-microbial abundance sponges [ 49 , 50 ], and Tethya sp. was only found at Upa-Upasina. Thirdly, the benthic holobiont community fraction was generated by scraping, and blending the scrapings from all 17 surfaces, including the remainder of the light-exposed top surface of the top plate. While this fraction will include both photosynthetic and non-photosynthetic organisms, the shaded plate surfaces constitute ~ 16 × the area of the light-exposed top surface of the top plate. The most abundant phyla found in this fraction on other reefs around the world are fairly consistent and include the Porifera, Cnidaria, Bryozoa, Chordata (Ascidiacea), and Annelida [ 42 , 43 , 44 ]. An example of an ARMS plate from which this fraction was collected can be seen in Fig.  2 E. This homogenised bulk sample was then subsampled (50 ml). Finally, the sediment fraction was generated by passing the water and sediment from the holding tank through a 500 μm and then a 100-μm sieve and collecting the material which did not pass through the 100-μm sieve. This drained sediment sample was subsampled (10 g). This fraction was therefore primarily composed of sediment, microbes, including free-living sediment dwelling microbes (e.g. bacteria) and microplankton (e.g. single-celled algae such as diatoms and dinoflagellates).

From each ARMS unit, each fraction was split in two; one-half was snap frozen for metabolomic analysis by being dropped in liquid nitrogen in a dry shipper, and the other was placed in RNA later for metabarcoding. All samples were returned to the Smithsonian National Museum of Natural History (Washington, DC, USA) in a liquid nitrogen dry shipper. Total DNA was extracted from 10 g of the benthic holobiont community samples and 5 g of the sediment samples using a MO-BIO PowerMax Soil DNA Isolation Kit according to the manufacturer’s protocol with the addition of 400 μg/ml proteinase K and an overnight lysis step at 56 °C and 200 rpm. The benthic photosynthetic community and sponge subsamples were extracted with the DNeasy Blood and Tissue Kit (Qiagen), according to the manufacturer’s instructions. All DNA extracts were purified using MO-BIO PowerClean DNA Clean-Up Kits, quantified Qubit dsDNA HS Kit, run on an agarose gel, and DNA quality investigated using ImageJ software. All sample types are known to contain relatively high bacterial biomass, and thus, we do not expect contamination from the lab environment or equipment to be a major issue in DNA libraries. However, extraction and PCR amplification controls were included for all sample types; these were all negative and so were not sequenced. Each sample was analysed with 16S rRNA gene metabarcoding (for the microbe community) and mass spectrometry (for metabolomics). All 16S rRNA gene libraries were prepared for sequencing using the original Earth Microbiome Project protocol using primers 515 F and 806 R, which are designed to amplify prokaryotes (Bacteria and Archaea) [ 51 ]. To investigate the benthic photosynthetic community, 23S rRNA gene libraries were also prepared using the protocol described by Marcelino and Verbruggen [ 52 ] using a two-step PCR procedure that first amplifies the gene fragment followed by ligation of the barcoded Illumina adaptors to the amplicons in a second PCR reaction; this protocol is designed to target both eukaryotic algae and cyanobacteria.

Metabolites were extracted from all fractions in 70% methanol [ 47 ]. The 70% methanol extraction was chosen to select for slightly polar molecules, encompassing a broad range of the chemosphere [ 53 ]. Metabolites were separated and identified via liquid chromatography-tandem mass spectrometry using a Bruker Daltonics Maxis qTOF mass spectrometer equipped with a standard electrospray ionisation source according to the methods of Quinn and colleagues [ 53 ]⁠. Briefly, the mobile phase was pumped through a Kinetex 2.6 μm C18 (30 × 2.10 mm) ultra-performance liquid chromatography (UPLC) column for a 15-min run. The resulting LC–MS/MS data files were processed through the MZmine2 workflow. The subsequent metabolite feature table was then processed through the GNPS feature-based molecular networking workflow with the default parameters, except that a minimum cosine of 0.65 and a minimum matched peaks of 4 were used for network construction.

Bioinformatics

A bioinformatic pipeline was implemented in R for the 16S and 23S rRNA gene libraries. Amplicon sequence variants (ASVs) were generated from raw sequencing data using the Divisive Amplicon Denoising Algorithm (DADA2 v1.24.0 [ 54 ]). Reads were quality filtered to maintain Q30 scores while maintaining at least 50 base pair overlap and removing any base pair below Q2 [ 55 ]. Default maxEE (2) and truncQ (2) parameters were used, 16S rRNA gene sequences were truncated at a length of 150 base pairs on both strands, 23S rRNA gene sequences had the first 20 base pairs (nonbiological primers) trimmed from both strands and were truncated at a length of 249 base pairs on the forward strand and 212 base pairs on the reverse strand (see Table S 1  for numbers of sequences passing denoising steps). Taxonomy was assigned using the DECIPHER v2.24.0 R package [ 56 ] — which has been shown to have higher accuracy than popular classifiers including BLAST and the RDP classifier [ 56 ] — and the GTDB (16S rRNA gene [ 57 ]) and microgreen (23S r RNA gene [ 58 ]) databases. 16S rRNA gene ASVs identified as plastids were subsequently removed. Samples were not rarefied as part of bioinformatic processing [ 59 ] but only when required for specific statistical analyses (see ASV-level Shannon diversity below).

Statistical analyses

PERMANOVAs were performed for each fraction separately, with the 16S rRNA gene sequencing and metabolomic data subdivided into community fractions (benthic photosynthetic community, benthic holobiont community and sediment) and organism fractions ( Halisarca sp. and Tethya sp. sponges), resulting in 10 PERMANOVAs (Table S 2 ). Prior to fitting PERMANOVAs, a multivariate analogue of Levene’s test for homogeneity of variances (betadisper) was applied to ensure PERMANOVA tests could be applied. PERMANOVAs fit locality and ordinal pH as explanatory variables, and locality was treated as a blocking factor, except in the case of the Tethya sp. sponge which was only found at one locality and so was fit with pH as the only explanatory variable. An eleventh PERMANOVA was run on the 23S rRNA gene data, following the same approach. Bonferroni corrections were applied to all p -values obtained from the PERMANOVAs to account for multiple testing. All PERMANOVAs and supporting NMDS visualisations were based on Morisita dissimilarities between samples, as they have been shown to be most reliable in the case of under sampling [ 60 ].⁠

Total ASV richness and phylum level Shannon diversity, each accounting for unobserved ASVs, were estimated for each metabarcoding sample using a breakaway model [ 61 ] and a DivNet model, treating all samples as independent observations, respectively [ 62 ]. The estimated richness and estimated Shannon diversity of all metabarcoding samples were then modelled using a single betta hierarchical mixed model for each metric (including all fractions). This modelling approach was chosen to account for explanatory variables, richness variance, and richness estimation error [ 61 ]. Fraction, ordinal pH, and the interaction of fraction and pH were included as fixed effects and locality as a random effect. Compound richness and Shannon diversity were calculated from untransformed data for metabolomic samples from all fractions. Each metric was modelled with a single linear mixed model. Fraction, pH, and the interaction of fraction and ordinal pH were included as fixed effects and locality as a random effect. In addition, ASV level Shannon diversity was calculated (no statistical estimation procedure was applied) for all samples rarefied to even depth ( n  = 50,000) to test the sensitivity of the results found for phylum level Shannon diversity.

The change in abundance of phyla and metabolites with OA was analysed using DeSeq2 differential abundance analysis with a negative binomial distribution [ 63 ]⁠. Differential abundance and dispersions were calculated for each community fraction (benthic photosynthetic community, benthic holobiont community, and sediment) and multiomic analysis type separately using a DESeq2 design formula with variables of locality and pH. This enabled change within each community fraction to be examined. However, abundance and dispersions were calculated for both sponge fractions together using a design formula with variables of species, locality, and pH. This enabled shared change occurring across sponges under OA to be examined. Wald significance tests were conducted for changes in differential abundance under OA, with a parametric fit of dispersions [ 63 ].

Microbiome/metabolome distinctness was calculated for each ARMS as the proportion of unique sequences found within the benthic holobiont community microbiome/metabolome which were not also found in the sediment microbiome/metabolome. Microbiome/metabolome distinctness was modelled using a linear mixed model with ordinal pH as a fixed effect and locality as a random effect. A likelihood ratio test was used to infer the significance of ordinal pH as a fixed effect. Note that this analysis was not conducted for the benthic photosynthetic community as we are testing whether distinctness is reduced for the general community of macro-organisms, and the benthic holobiont community is a more general sample including both photosynthetic and non-photosynthetic organisms. The same approach was taken to calculate and model microbiome/metabolome distinctness for individual holobionts with the additional random effects of (i) ARMS identity (nested within locality), as multiple individual holobionts were collected from the same ARMS, and (ii) sponge species.

Benthic holobiont community microbiome distinctness from the sediment microbiome was also modelled with a modified mixture Sloan neutral community model (MSNCM [ 64 ]). This additional modelling approach captures the contribution of each of the sediment and benthic holobiont community microbiome metacommunities to the composition of benthic holobiont community microbiomes from individual ARMS, thus providing an alternative abundance-based test of whether benthic holobiont community microbiomes become more distinct from the sediment microbiome under OA. The original Sloan neutral community model describes the frequency of occurrence of ASVs in a community as a function of their abundance in the metacommunity, with a single free parameter (m: migration) which can be interpreted as the probability of neutral dispersal or alternatively inverse dispersal limitation. The MSNCM used here models ASV frequency in sampled benthic holobiont community microbiomes from each pH regime as a function of its abundance in two metacommunities: (1) all benthic holobiont community microbiomes from the same pH regime as the sample and (2) all sediment microbiomes from the same pH regime as the sample. Each metacommunity is fit with its own migration/inverse dispersal limitation parameter (mholo, menv), and a mixture parameter (mix) is fit describing the contribution of each metacommunity. The model is fit to samples from each pH regime separately using non-linear least-squares fitting as detailed in Burns et al. [ 65 ]⁠.

All statistical analyses were conducted in R (version 4.2.1 [ 66 ])⁠; specific packages used were as follows: phyloseq v1.40.0 [ 67 ] for data manipulation, vegan v2.6–4 [ 68 ] for PERMANOVA and betadisper, breakaway v4.8.2 [ 61 ] and DivNet v.0.4.0 [ 62 ] for diversity estimation and modelling, lme4 v1.1–31 [ 69 ] and MuMIn v1.47.1 [ 70 ] for generalised linear mixed models, DESeq2 v1.36.0 for differential abundance analysis [ 63 ], and minpack.lm v1.2–2 [ 71 ] and Hmisc v4.7–1 [ 72 ] for non-linear least-squares modelling.

Genetic diversity and composition

Five fractions were generated for analysis: benthic photosynthetic community, benthic holobiont community, sediment, Tethya sp., and Halisarca sp. Benthic photosynthetic communities were dominated by red algae (Rhodophyta), brown algae (Ochrophyta), and Cyanobacteria. Benthic holobiont communities were visually dominated by Porifera, Chordata, Bryozoa, Annelida, Arthropoda, and Mollusca. Benthic holobiont community microbiomes were dominated by Proteobacteria, unclassified Bacteria, and Cyanobacteria. Benthic photosynthetic community microbiomes were dominated by Proteobacteria and Cyanobacteria, followed by Firmicutes, Bacteroidota, and unclassified Bacteria. Sediment microbiomes were dominated by Proteobacteria, unclassified Bacteria, Bacteroidota, and Planctomycetota. Sponge microbiomes were dominated by Proteobacteria; unclassified Bacteria ASVs were also highly abundant in Tethya sp. samples. See Figure S 1 .

Fifteen 23S rRNA gene metabarcoding libraries were generated across the pH gradient, to confirm the expected effect of OA on the benthic photosynthetic communities. The composition of the benthic photosynthetic community differed significantly by pH ( F  = 5.5, p  < 0.05), with significant declines in phylum Shannon diversity (95% CI [–0.76, –0.55], p  < 0.05) and ASV Shannon diversity (95% CI [–2.27, –0.33], p  < 0.05) at lower pH. Lower pH was associated with significantly increased differential abundance of the dominant phylum Ochrophyta (of which 99.7% of reads were from the class Phaeophyceae, and 71.4% were from the genus Sargassum ; Fig.  3 ; Figure S 1 ).

Heat maps of significant differential abundance of phyla with decreasing pH. Significant change in differential abundance of algal phyla (23S rRNA gene) with decreasing pH are seen in ( A ); algal phyla are shown on the left, and families are shown on the right. Algal taxonomy was assigned using the microgreen database [ 58 ]. Significant change in differential abundance of microbial phyla (16S rRNA gene) with decreasing pH are seen in ( B ); bacterial phyla are shown on the left. Microbial taxonomy was assigned using the GTDB taxonomy [ 57 ]

Ninety-four 16S rRNA metabarcoding libraries were generated across the pH gradient, from 18 ARMS. Each fraction (benthic photosynthetic community microbiome, sediment microbiome, benthic holobiont community microbiome, and sponge [ Tethya sp. and Halisarca sp.] microbiomes) had between 15 and 30 samples (Table 1 ), which in total produced 55,348 ASVs ( n  = 94). Eighty bacterial phyla were identified, with 77.7% of reads identified to the level of phylum.

Forty-seven 16S rRNA gene metabarcoding libraries were generated across the benthic photosynthetic community, benthic holobiont community, and sediment microbiomes ( n  = 15, 17, 15, respectively). All community microbiomes were significantly compositional different at lower pH (control pH compared with medium and low pH as an ordinal variable): benthic photosynthetic community microbiome ( F  = 2.3, p  < 0.05), sediment microbiome ( F  = 3.9, p  < 0.05), and benthic holobiont community microbiome ( F  = 3.0, p  < 0.05; Table S 2 B; Figure S 2 ). There was no significant effect of pH on richness for any community microbiome (Table S 3 ). Phylum and ASV level Shannon diversity was significantly lower in the benthic photosynthetic community microbiome at lower pH (95% CI [− 0.55, − 0.09], p  < 0.05; Table S 4 ; Figure S 3 and 95% CI [− 2.5, − 0.64], p  < 0.05; Table S 5 ; Figure S 3 , respectively). Please see Table S 6 for a summary of all significant 16S rRNA patterns. Decreased pH was associated with significant differences in the abundance of the following phyla: increased WOR-3 and Desulfobacterota alongside decreased Armatimonadota in the sediment microbiome, increased Desulfobacterota in the benthic holobiont community microbiome, and increased Desulfobacterota alongside decreased Poribacteria, Gemmatimonadota, and Firmicutes in the benthic photosynthetic community microbiome (Fig.  3 ).

Forty-seven 16S rRNA gene metabarcoding libraries were generated from the microbiomes of the two sponge species: 17 from Halisarca sp. individuals and 30 from Tethya sp. individuals. Microbiome composition was only significantly different across the pH gradient for Tethya sp. sponges ( F  = 5.9, p  < 0.05; Table S 2 ; Figure S 2 ). While there was no significant effect of pH on ASV richness or ASV level Shannon diversity, reduced pH was associated with significantly lower phylum level Shannon diversity in Tethya sp. sponge microbiomes (95% CI [− 0.44, − 0.12], p  < 0.05), but no such effect was found in Halisarca sp. sponge microbiomes (Table S 4 ; Figure S 3 ; see Table S 6 for a summary of all significant 16S rRNA patterns). At the phyla level, lower pH was associated with significantly higher abundances of Marinisomatota, SAR324 , and Cyanobacteria and significantly lower abundances of Firmicutes, Desulfobacterota, and Bacteroidota (Fig.  3 ) in both sponge species microbiomes.

Biochemical diversity and composition

One-hundred and seven metabolome libraries were analysed from the same 18 ARMS. Each fraction (benthic photosynthetic community metabolome; benthic holobiont community metabolome, sediment metabolome, and sponge metabolomes) had between 18 and 34 samples across the pH gradient (Table 1 ). These samples produced 1211 compounds, of which 4.62% were identified, representing 6% of all molecules.

Fifty-five metabolome libraries were generated from the sediment metabolome, the benthic holobiont community, and the benthic photosynthetic community metabolomes ( n  = 18, 18, 19, respectively). There was no significant compositional difference in these community metabolomes at lower pH (Table S 2 ; Figure S 2 ). There was also no significant effect of pH on compound richness for any community metabolome (Table S 7 ; Figure S 3 ). However, Shannon diversity was significantly lower at lower pH in the benthic holobiont community metabolome (95% CI [− 0.07, − 1.18], p  < 0.05) and significantly higher at lower pH in the sediment metabolome (95% CI [0.02, 0.80], p  < 0.05; Table S 8 ; Figure S 3 ). Decreased pH was associated with significant differences in the abundance of several identified compounds (note that more than 95% of compounds were not identifiable). In the sediment metabolome, glycerophospholipids (lysophosphatidylcholines (LPCs) and phosphocholines) and pheophorbide A (a chlorophyll-derived compound) were less abundant, and beta-carotene was more abundant. In the benthic holobiont community benzene derivatives, chondramide B and mesoporphyrin IX (the latter two both anticarcinogens) were less abundant; a range of glycerophospholipids (again LPCs and phosphocholines), sucrose, and beta-carotene were more abundant. In the benthic photosynthetic community, pheophorbide A was more abundant, and various glycerophospholipids had significantly different abundances, with LPCs mostly having higher abundances and phosphocholines lower abundances (Fig.  4 ).

Heat map of significant differential abundance of metabolites with decreasing pH (community and sponge). Identities of metabolites 1–29 are outlined below: 01 — (1S,2S)-2-(methylamino)-1-phenylpropan-1-ol hydrochloride; 02 — benzalkonium chloride (C12); 03 — diphenhydramine|2-benzhydryloxy-N,N-dimethylethanamine; 04 — N,N-diethyl-3-methylbenzamide; 05 — niranthin; 06 — beta-carotene; 07 — chondramide B; 08 — 15(S)-hydroxy-(5Z,8Z,11Z,13E)-eicosatetraenoic acid; 09 — 17(18)-EpETE; 10 — 1-octadecyl-sn-glycero-3-phosphocholine; 11 — 1-palmitoylphosphatidylcholine; 12 — 1-(1Z-hexadecenyl)-sn-glycero-3-phosphocholine; 13 — 1-(9Z-Octadecenoyl)-sn-glycero-3-phosphocholine; 14 — 1-arachidoyl-2-hydroxy-sn-glycero-3-phosphocholine; 15 — 1-hexadecanoyl-sn-glycero-3-phosphocholine; 16 — 1-O-hexadecyl-2-O-(2E-butenoyl)-sn-glyceryl-3-phosphocholine; 17 — 1-octadecyl-sn-glycero-3-phosphocholine; 18 — 1-stearoyl-2-hydroxy-sn-glycero-3-phosphocholine; 19 — lysophosphatidylcholine (LPC 16:0); 20 — lysophosphatidylcholine (LPC 18:1); 21 — lysophosphatidylcholine (LPC 18:2); 22 — lysophosphatidylcholine (LPC 18:3); 23 — lysophosphatidylcholine (LPC 20:5); 24 — lysophosphatidylcholine (LPC 22:6); 25 — mesoporphyrin IX; 26 — 3-indoleacrylic acid; 27 — sucrose; 28 — (R)-4-((3R,5R,8R,9S,10S,12S,13R,14S,17R)-3,12-dihydroxy-10,13-dimethylhexadecahydro-1H-cyclopenta[a]phenanthren-17-yl)pent-2-enoic acid; 29 — pheophorbide A

Fifty-two metabolome libraries were generated from the two sponge species, with 18 from Halisarca sp. and 34 from Tethya sp. There was no significant compositional difference in the metabolomes of either sponge at different pH (Table S 2 ; Figure S 2 ). There was also no significant effect of pH on compound richness or Shannon diversity for the sponge metabolomes (Figure S 3 ). Among identified compounds (note that more than 95% of compounds were not identifiable), decreased pH was associated with significantly different abundances in a variety of glycerophospholipids and a significantly increased abundance of a benzene derivate across both sponge species (Fig.  4 ).

Holobiont microbial and chemical distinctness

The proportion of benthic holobiont community microbiome ASVs not found in the sediment microbiome (i.e. holobiont community microbiome distinctness) was lower at lower pH (95% CI [− 6.36, − 1.74], p  < 0.05). The same pattern was observed in the metabolome: the proportion of benthic holobiont community metabolites not found in the sediment (i.e. holobiont community metabolome distinctness) was lower at lower pH (95% CI [− 14.46, − 2.83], p  < 0.05; Fig.  5 ).

Microbiome and metabolome distinctness of benthic holobiont communities as a function of pH. Microbiome ( A ) and metabolome ( B ) of the benthic holobiont community versus the sediment microbiome and microbiome ( C ) and metabolome ( D ) of the two sponge microbiomes versus the sediment microbiome. Distinctness was calculated as percentage of ASVs/metabolites not shared between microbiomes/metabolomes. Horizontal dotted line indicates 50% distinct. ** p -value < 0.01

The dispersal probability from the sediment microbiome into the benthic holobiont community microbiome, as measured by the MSNCM (menv weighted by the mixing parameter), was also higher at lower pH. The frequency of occurrence of ASVs across the benthic holobiont community microbiome samples was well described by their abundance in the benthic holobiont community and sediment metacommunities through the MSNCM. However, the fit of the model was poorer at medium and low pH (control: 0.64, medium: 0.29, low: 0.38). The ratio between mholo and menv (both weighted by the mixing parameter) was also lower at lower pH, reflecting an overall increased contribution of sediment microbial abundance to determining the microbiome composition of benthic holobiont communities under OA (Table 2 ).

Change in microbial and chemical distinctness of individual sponge holobionts ( Tethya sp. and Halisarca sp.) with ocean acidification was also analysed to examine whether community-level patterns were observed in individual holobionts. Individual sponge holobiont microbiome distinctness from sediment was lower at lower pH (95% CI [− 7.87, − 1.75], p  < 0.05; Fig. 5 ). This individual holobiont microbiome distinctness model explained 61% of variation ( R 2 ) compared to 9% of variation ( R 2 ) explained by the equivalent community-level microbiome distinctness model described above. There was no significant effect of pH on sponge metabolome distinctness.

Our results are consistent with the simplification of the algal community seen under OA elsewhere [ 73 ]. Our results further demonstrate that this simplification effect extends across the entire benthic photosynthetic community holobiome as algal-associated microbes also decline in Shannon diversity with OA. While we do not observe an overall compositional shift in the benthic photosynthetic community metabolome, implying that photosynthetic function is largely conserved through these changes, we do see shifts in specific metabolites. Increasing concentration of chlorophyll-derived pheophorbide A suggests increased chlorophyll turnover [ 74 ], consistent with the doubling of macroalgal benthic cover previously observed under OA at these sites [ 18 ]. We also see significant increases in sucrose in the benthic holobiont community under OA, potentially related to sugar-enriched dissolved organic carbon (DOC) released by these algae.

Both findings are consistent with the initial steps in the DDAM mechanism of microbialisation [ 26 ], whereby increased DOC is released from fleshy algae, in this case Sargassum [ 19 ]. We did not, however, test for the increased microbial biomass previously shown to result from increased DOC stimulating the microbial loop and causing a shift in ecosystem trophic structure [ 26 , 75 ]. The main microbial phyla that we observed to increase in abundance in our community samples (i.e. not sponge samples) with low pH (Desulfobacterota and WOR-3 ) differ from those observed by Haas [ 26 ] (Alphaproteobacteria, Gammaproteobacteria, and Bacteroidetes). However, this difference in specific taxonomic patterns in response to different stressors, across different sites, and when sampling different substrates (water vs. sediment) is expected and is a key motivation for seeking to identify more general ecosystem-level effects of stressors, such as microbiome distinctness.

The data presented here provide the first evidence of declining holobiont community distinctness in microbes and metabolites under OA. Our results build on the evidence that OA changes holobiont microbiomes [ 76 ] by demonstrating a systematic decline of a distinct benthic holobiont community microbiome, such that it becomes more compositionally similar to the sediment microbiome. The composition of the sediment microbiome in our study is representative of surface sediments from the region [ 77 ] and global analyses of marine sediments [ 78 ]. Specifically, the results of the MSNCM suggest that this effect may result from the benthic holobiont community being increasingly colonised by sediment microbes as pH decreases. A similar effect has previously been observed for specific sponge [ 79 ] and coral [ 80 , 81 ] holobionts in response to different environmental stressors, but has not previously been observed at the community level. While organism-level studies provide information about the response of key species (e.g. habitat builders) to environmental change, a holistic approach is needed to accurately evaluate and predict impacts on coral reefs. The synthesis of knowledge across scales, from individual microbes and holobionts to ecosystem-wide communities and processes, has recently been called for by multiple authors [ 29 , 75 , 82 ]. Autonomous reef monitoring structures (ARMS) provide a novel tool for taking this “nested ecosystem approach” and conducting in situ experiments.

We explain the decline in the distinctness of the benthic holobiont community from the sediment microbiome as being caused by increased opportunities for colonisation of benthic holobiont communities by environmental microbes due to microbialisation. However, individual macrobes under stress may also become less able to regulate their microbiomes, while colonisation opportunities remain constant. In extreme cases, this inability to regulate the microbiome can result in traumatic dysbiosis [ 80 ], a more heterogeneous microbiome (Anna Karenina principle, [ 83 ], and host death. However, it is unlikely that this process is the major contributor to the observed community-level effect seen here because macrobe community compositions have already shifted under OA conditions at these vent sites, with an increased dominance of taxa that are less impacted by the stressor [ 18 , 84 ]. Therefore, the notion that the majority of the macrobe community is experiencing dysbiosis associated with acute organism-level stress seems unlikely. For example, some sponges are known to thrive under OA, and do not exhibit evidence of organismal stress [ 85 , 86 , 87 ]. In this study, the two sponge holobionts individually analysed ( Tethya sp. and Halisarca sp.) showed reduced distinctness of their microbiomes from the sediment microbiome under OA. However, neither showed evidence of increased compositional heterogeneity of their microbiomes as expected under dysbiosis by the Anna Karenina principle, which predicts organism-level stress to reduce the ability of macrobes to regulate their microbiomes [ 83 ]. In addition, metabolomes for these two sponges do not become significantly less distinct from the sediment under OA, which is consistent with the hypothesis that these sponges were not under stress.

Colonisation of holobionts by environmental microbes may support the resilience of macrobe communities, as it has been shown to allow some hosts to acclimatise to new environmental conditions, for example by allowing the host to make use of changing energy sources, and facilitate greater adaptation than can be afforded by host phenotypic plasticity [ 88 , 89 ]. Different degrees of microbial restructuring observed among different sponge species indicate that horizontal transmission differs between species, and these variations affect the ability of sponges to persist under OA conditions [ 12 , 90 ]. Here, we find significant increases in Cyanobacteria associated with sponges under OA conditions, which can contribute > 50% of a sponge’s carbon demand [ 91 ] and likely provide at least some sponge species with enhanced scope for growth in these seep environments [ 12 ]. We also find significant increases in Desulfobacterota in the benthic photosynthetic community, benthic holobiont community, and sediment; this phylum includes many organisms capable of reducing sulphur compounds [ 92 , 93 ]. Only one of the two ocean vents (Dobu) is known to release hydrogen sulphide [ 18 ], so this does not explain the increase in Desulfobacterota observed across sites. We note that while we do not identify changes in the differential abundance of any metabolites with known roles in sulphur cycling, this is not evidence of their absence due to the low level (less than 5%) of identification of metabolites. While their role here is unknown, coral reefs are important hotspots of marine sulphur and increased sulphate reduction rates of marine microbial communities have been found to occur between a pH of 6 and 7 [ 94 ], suggesting that rates may increase with OA. As the marine sulphur cycle is a quintessential example of algal–bacterial interactions [ 95 ], it will be important for future studies to investigate the impact of algal-derived microbialisation on the marine sulphur cycle, especially as new components and pathways in the sulphur cycle are still being identified [ 96 ].

One would expect the dynamics of microbes and holobionts to be universal to all ecosystems [ 36 , 97 , 98 ], though they may emerge from different organism-level interactions. Therefore, microbialisation, and the observable property of declining holobiont community distinctness under environmental change, could represent a universal ecosystem stress response. Identifying such a general, undesirable response (microbialised ecosystems typically have lower intrinsic and use values [ 99 ]) and a metric of ecosystem change has clear benefits to policy and evaluation. For example, ecosystem change and the associated risk of ecosystem collapse are the underpinning concept leveraged for the IUCN Red List of Ecosystems [ 100 ], but defining collapse for each ecosystem individually is a time-consuming and contentiously value-laden task [ 101 , 102 ]. Furthermore, as microbial communities respond rapidly to environmental change, microbial bioindicators could provide signatures of change with the speed and resolution to allow real-time responses by ecosystem managers. Generating predictions of ecosystem change based on a mechanistic understanding of all organism-level effects of stressors remains unrealistic [ 17 , 103 ]. Therefore, identifying general ecosystem-level changes under stress presents a promising route towards a more efficient predictive ecosystem science, responding to the urgent needs of the biodiversity and climate crisis.

Availability of data and materials

The metabarcoding datasets are available from the NCBI Sequence Read Archive repository. 16S rRNA gene data is available under BioProject ID PRJNA945340 ( http://www.ncbi.nlm.nih.gov/bioproject/945340 ) and 23S rRNA gene data is available under BioProject ID PRJNA945259 ( http://www.ncbi.nlm.nih.gov/bioproject/945259 ). The mass spectrometry data is available at the GNPS MassIVE repository under MassIVE ID: MSV000080572 ( https://gnps.ucsd.edu/ProteoSAFe/status.jsp?task=f5c6591769d541a68fdb8bb201532054 ). R scripts used in the bioinformatic pipeline are archived at https://doi.org/10.5281/zenodo.7740559 and available at https://github.com/J-Cos/BioinformaticPipeline . R scripts for the statistical analysis are archived at https://doi.org/10.5281/zenodo.8280507 and available at: https://github.com/J-Cos/Paper_PNG .

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Acknowledgements

We thank Sam Noonan, Sven Uthicke, Craig Humphrey, Amanda Feuerstein, and Obedi Daniel, along with Rob van der Loos and crew members of the M/V Chertan, for their help and support in the field. We also thank the community of Upa Upasina and Dobu Island for permission to deploy ARMS on their reefs and for their generous welcome and assistance during the over 2 years of this experiment. We wish to acknowledge the use of facilities and technical support from the Laboratories of Analytical Biology, National Museum of Natural History, Smithsonian Institution, and the Smithsonian Institution’s DNA Barcode Network in particular Lee Weigt, Amy Driskell, Jeff Hunt, Lowen Wachhaus, Matthew Kweskin, Maggie Halloran, and Janette Madera as well as Mike Trizna and Niamh Redmond. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the US government.

JW was supported by the QMEE CDT and funded by NERC grant number NE/R012229/1; CM and ER were supported by the National Science Foundaion (Award Number 1243541); NK and LP were supported by the National Science Foundation (Award Number 1558868), the Sant Chair for Marine Science, and the Smithsonian Institution’s Scholarly Studies Program; AH was supported by the National Science Foundation (Award Number 2022717); KF was supported by the Great Barrier Reef Foundation’s ‘Resilient Coral Reefs Successfully Adapting to Climate Change’ programme.

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Georgina Mace Centre for the Living Planet, Department of Life Sciences, Imperial College London, Buckhurst Road, Ascot, SL5 7PY, UK

Jake Williams & Emma Ransome

Institute of Zoology, Zoological Society of London, Regent’s Park, London, NW1 4RY, UK

Jake Williams & Nathalie Pettorelli

Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, USA

Aaron C. Hartmann

Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI, 48824, USA

Robert A. Quinn

Laboratoire Evolution Et Diversité Biologique, CNRS/UPS, Toulouse, France

Laetitia Plaisance

National Museum of Natural History, Smithsonian Institution, Washington, DC, 20013, USA

Laetitia Plaisance, Michael O’Mahoney, Chris P. Meyer & Nancy Knowlton

Australian Institute of Marine Science, Townsville, Queensland, Australia

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ER & LP designed the experiment; ER & LP collected the data; ER & MO conducted the metabarcoding; RQ conducted the mass spectrometry; RO & AH ran the metabolomics, JW ran the bioinformatics; JW & ER conceived the hypotheses, JW analysed the results, JW wrote the paper. All authors contributed significantly to editing the manuscript.

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Additional file 1:.

  Table S1 . Number of sequences retained at each step of denoising (implemented in DADA2); samples with fewer than 50,000 denoised sequences (ASVs) and duplicate samples were removed prior to analysis. Table S2 . Effect size and significance of factors in PERMANOVAs describing the compositional differences among sites, and along the pH gradient for all multiomic data types; site fit as a blocking factor, except in the case of Halisarca sp. sponge fractions, which were only collected at one site. A p -value less than 0.05 indicates that the factor significantly affects composition. Bonferroni correction was applied to all PERMANOVA. Table S3 . ASV estimated richness betta mixed model: fixed effects listed, random effect of locality. Model Explanatory Power: test statistic = 117.2, p <0.05. A p -value less than 0.05 indicates that the explanatory variable significantly affects ASV richness. All values in the table are reported to two significant figures. Table S4. Phylum Shannon diversity betta mixed model: fixed effects listed, random effect of locality. Model Explanatory Power: test statistic = 3197.94, p <0.05. A p-value less than 0.05 indicates that the explanatory variable significantly affects metabolite richness. All values in the table are reported to two significant figures. Table S5. ASV Shannon diversity betta mixed model: fixed effects listed, random effect of locality. R Squared (conditional)= 88.8%. A p -value less than 0.05 indicates that the explanatory variable significantly affects metabolite richness. All values in the table are reported to two significant figures. Table S6. Summary of tests and results for 16S rRNA gene and metabolomic data. Table S7. Compound richness linear mixed model: fixed effects listed, random effect of locality. R Squared (conditional)= 24.5%. A p -value less than 0.05 indicates that the explanatory variable significantly affects ASV richness. All values in the table are reported to two significant figures. Table S8. Compound Shannon diversity linear mixed model: fixed effects listed, random effect of locality. R Squared (conditional)= 32.0%. A p -value less than 0.05 indicates that the explanatory variable significantly affects ASV richness. All values in the table are reported to two significant figures.  Fig. S1. Visual representation of read abundance from the ARMS 23S rRNA gene and 16S rRNA gene metabarcoding dataset. Data are aggregated across biological replicates to present the average composition of each fraction, at each pH, showing phylum (in white text) and class (in grey text). Fig. S2. NMDS of microbiome and metabolome composition of all fractions across the pH gradient, calculated using Morisita dissimilarity. Fig. S3. Microbial and chemical richness and Shannon diversity boxplots for all fractions across the pH gradient.

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Williams, J., Pettorelli, N., Hartmann, A.C. et al. Decline of a distinct coral reef holobiont community under ocean acidification. Microbiome 12 , 75 (2024). https://doi.org/10.1186/s40168-023-01683-y

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Received : 16 March 2023

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DOI : https://doi.org/10.1186/s40168-023-01683-y

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