hypothesis of a given conditional statement

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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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.

About Cuemath

At  Cuemath , our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

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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2.4 Truth Tables for the Conditional and Biconditional

Learning objectives.

After completing this section, you should be able to:

• Use and apply the conditional to construct a truth table.
• Use and apply the biconditional to construct a truth table.
• Use truth tables to determine the validity of conditional and biconditional statements.

Computer languages use if-then or if-then-else statements as decision statements:

• If the hypothesis is true, then do something.
• Or, if the hypothesis is true, then do something; else do something else.

For example, the following representation of computer code creates an if-then-else decision statement:

Check value of variable i i .

If i < 1 i < 1 , then print "Hello, World!" else print "Goodbye".

In this imaginary program, the if-then statement evaluates and acts on the value of the variable i i . For instance, if i = 0 i = 0 , the program would consider the statement i < 1 i < 1 as true and “Hello, World!” would appear on the computer screen. If instead, i = 3 i = 3 , the program would consider the statement i < 1 i < 1 as false (because 3 is greater than 1), and print “Goodbye” on the screen.

In this section, we will apply similar reasoning without the use of computer programs.

People in Mathematics

The Countess of Lovelace, Ada Lovelace, is credited with writing the first computer program. She wrote an algorithm to work with Charles Babbage’s Analytical Engine that could compute the Bernoulli numbers in 1843. In doing so, she became the first person to write a program for a machine that would produce more than just a simple calculation. The computer programming language ADA is named after her.

Reference: Posamentier, Alfred and Spreitzer Christian, “Chapter 34 Ada Lovelace: English (1815-1852)” pp. 272-278, Math Makers: The Lives and Works of 50 Famous Mathematicians , Prometheus Books, 2019.

Use and Apply the Conditional to Construct a Truth Table

A conditional is a logical statement of the form if p p , then q q . The conditional statement in logic is a promise or contract. The only time the conditional, p → q , p → q , is false is when the contract or promise is broken.

For example, consider the following scenario. A child’s parent says, “If you do your homework, then you can play your video games.” The child really wants to play their video games, so they get started right away, finish within an hour, and then show their parent the completed homework. The parent thanks the child for doing a great job on their homework and allows them to play video games. Both the parent and child are happy. The contract was satisfied; true implies true is true.

Now, suppose the child does not start their homework right away, and then struggles to complete it. They eventually finish and show it to their parent. The parent again thanks the child for completing their homework, but then informs the child that it is too late in the evening to play video games, and that they must begin to get ready for bed. Now, the child is really upset. They held up their part of the contract, but they did not receive the promised reward. The contract was broken; true implies false is false.

So, what happens if the child does not do their homework? In this case, the hypothesis is false. No contract has been entered, therefore, no contract can be broken. If the conclusion is false, the child does not get to play video games and might not be happy, but this outcome is expected because the child did not complete their end of the bargain. They did not complete their homework. False implies false is true. The last option is not as intuitive. If the parent lets the child play video games, even if they did not do their homework, neither parent nor child are going to be upset. False implies true is true.

The truth table for the conditional statement below summarizes these results.

Notice that the only time the conditional statement, p → q , p → q , is false is when the hypothesis, p p , is true and the conclusion, q q , is false.

Logic Part 8: The Conditional and Tautologies

Example 2.18

Constructing truth tables for conditional statements.

Assume both of the following statements are true: p p : My sibling washed the dishes, and q q : My parents paid them $5.00. Create a truth table to determine the truth value of each of the following conditional statements.

• p → q p → q
• p → ~ q p → ~ q
• ~ p → q ~ p → q

Your Turn 2.18

Example 2.19, determining validity of conditional statements.

Construct a truth table to analyze all possible outcomes for each of the following statements then determine whether they are valid.

• p ∧ q → ~ q p ∧ q → ~ q
• p → ~ p ∨ q p → ~ p ∨ q

Your Turn 2.19

Use and apply the biconditional to construct a truth table.

The biconditional, p ↔ q p ↔ q , is a two way contract; it is equivalent to the statement ( p → q ) ∧ ( q → p ) . ( p → q ) ∧ ( q → p ) . A biconditional statement, p ↔ q , p ↔ q , is true whenever the truth value of the hypothesis matches the truth value of the conclusion, otherwise it is false.

The truth table for the biconditional is summarized below.

Logic Part 11B: Biconditional and Summary of Truth Value Rules in Logic

Example 2.20

Constructing truth tables for biconditional statements.

Assume both of the following statements are true: p p : The plumber fixed the leak, and q q : The homeowner paid the plumber $150.00. Create a truth table to determine the truth value of each of the following biconditional statements.

• p ↔ q p ↔ q
• p ↔ ~ q p ↔ ~ q
• ~ p ↔ ~ q ~ p ↔ ~ q

Your Turn 2.20

The biconditional, p ↔ q , p ↔ q , is true whenever the truth values of p p and q q match, otherwise it is false.

Logic Part 13: Truth Tables to Determine if Argument is Valid or Invalid

Example 2.21

Determining validity of biconditional statements.

Construct a truth table to analyze all possible outcomes for each of the following statements, then determine whether they are valid.

• p ∧ q ↔ p ∧ ~ q p ∧ q ↔ p ∧ ~ q
• p ∨ q ↔ ~ p ∨ q p ∨ q ↔ ~ p ∨ q
• p → q ↔ ~ q → ~ p p → q ↔ ~ q → ~ p
• p ∧ q → ~ r ↔ p ∧ q ∧ r p ∧ q → ~ r ↔ p ∧ q ∧ r

Your Turn 2.21

Check your understanding, section 2.4 exercises.

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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.

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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

Get help learning about conditional statements

Understanding conditional statements can be tricky, especially when it gets deeper into programming language. If your student needs a boost in their comprehension of conditional statements, have them meet with an expert tutor who can give them 1-on-1 support in a setting free from distractions. A tutor can work at your student's pace so that tutoring is efficient while using their learning style - so that tutoring is effective. To learn more about how tutoring can help your student master conditional statements, contact the Educational Directors at Varsity Tutors today.

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2.12: Converse, Inverse, and Contrapositive Statements

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Conditional statements drawn from an if-then statement.

Converse, Inverse, and Contrapositive

Consider the statement: If the weather is nice, then I’ll wash the car. We can rewrite this statement using letters to represent the hypothesis and conclusion.

\(p=the\: weather \:is \:nice \qquad q=I'll \:wash \:the \:car\)

Now the statement is: if \(p\), then \(q\), which can also be written as \(p\rightarrow q\).

We can also make the negations, or “nots,” of \(p\) and \(q\). The symbolic version of "not p" is \(\sim p.

\(\sim p=the \:weather \:is \:not \:nice \qquad \sim q=I \:won't \:wash \:the \:car\)

Using these “nots” and switching the order of \(p\) and \(q\), we can create three new statements.

\(Converse \qquad q\rightarrow p \qquad \underbrace{If\: I\: wash\: the\: car}_\text{q}, \underbrace{then\: the \:weather \:is \: nice}_\text{p}\).

\(Inverse \qquad \sim p\rightarrow \sim q \qquad \underbrace{If\: the\: weather\: is \:not \:nice}_\text{p}, \underbrace{\:then \:I \:won't \:wash \:the \:car}_\text{q}\).

\(Contrapositive \qquad \sim q\rightarrow \sim p \qquad \underbrace{If\: I \:don't \:wash \:the \:car}_\text{q}, \underbrace{then the weather is not nice}_\text{p}\).

If the “if-then” statement is true, then the contrapositive is also true. The contrapositive is logically equivalent to the original statement. The converse and inverse may or may not be true. When the original statement and converse are both true then the statement is a biconditional statement . In other words, if \(p\rightarrow q\) is true and \(q\rightarrow p\) is true, then \(p \leftrightarrow q\) (said “\(p\) if and only if \(q\)”).

What if you were given a conditional statement like "If I walk to school, then I will be late"? How could you rearrange and/or negate this statement to form new conditional statements?

Example \(\PageIndex{1}\)

If \(n>2\), then \(n^{2}>4\).

Find the converse, inverse, and contrapositive. Determine if each resulting statement is true or false. If it is false, find a counterexample.

The original statement is true.

\(\underline{Converse}\): If \(n^{2}>4\), then \(n>2\).

False. If \(n^{2}=9\), \(n=−3\: or \: 3\). \((−3)^{2}=9\)

\(\underline{Inverse}\): If \(n\leq 2\), then \(n^{2}\leq 4\).

False. If \(n=−3\), then \(n^{2}=9\).

\(\underline{Contrapositive}\): If \(n^{2}\leq 4\), then \(n\leq 2\).

True. The only \(n^{2}\leq 4\) is 1 or 4. \(\sqrt{1}=\pm 1\) and\(\sqrt{4}=\pm 2\), which are both less than or equal to 2.

Example \(\PageIndex{2}\)

If I am at Disneyland, then I am in California.

\(\underline{Converse}\): If I am in California, then I am at Disneyland.

False. I could be in San Francisco.

\(\underline{Inverse}\): If I am not at Disneyland, then I am not in California.

False. Again, I could be in San Francisco.

\(\underline{Contrapositive}\): If I am not in California, then I am not at Disneyland.

True. If I am not in the state, I couldn't be at Disneyland.

Notice for the converse and inverse we can use the same counterexample.

Example \(\PageIndex{3}\)

Rewrite as a biconditional statement: Any two points are collinear.

This statement can be rewritten as:

Two points are on the same line if and only if they are collinear. Replace the “if-then” with “if and only if” in the middle of the statement.

Example \(\PageIndex{4}\)

Any two points are collinear.

First, change the statement into an “if-then” statement:

If two points are on the same line, then they are collinear.

\(\underline{Converse}\): If two points are collinear, then they are on the same line. True.

\(\underline{Inverse}\): If two points are not on the same line, then they are not collinear. True.

\(\underline{Contrapositive}\): If two points are not collinear, then they do not lie on the same line. True.

Example \(\PageIndex{5}\)

The following is a true statement:

\(m\angle ABC>90^{\circ}\) if and only if \(\angle ABC\) is an obtuse angle.

Determine the two true statements within this biconditional.

Statement 1: If \(m\angle ABC>90^{\circ}\), then \(\angle ABC\) is an obtuse angle.

Statement 2: If \(\angle ABC\) is an obtuse angle, then \(m\angle ABC>90^{\circ}\).

For questions 1-4, use the statement:

If \(AB=5\) and \(BC=5\), then \(B\) is the midpoint of \(\overline{AC}\).

• Is this a true statement? If not, what is a counterexample?
• Find the converse of this statement. Is it true?
• Find the inverse of this statement. Is it true?
• Find the contrapositive of this statement. Which statement is it the same as?

Find the converse of each true if-then statement. If the converse is true, write the biconditional statement.

• An acute angle is less than \(90^{\circ}\).
• If you are at the beach, then you are sun burnt.
• If \(x>4\), then \(x+3>7\).

For questions 8-10, determine the two true conditional statements from the given biconditional statements.

• A U.S. citizen can vote if and only if he or she is 18 or more years old.
• A whole number is prime if and only if its factors are 1 and itself.
• \(2x=18\) if and only if \(x=9\).

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.4.

Additional Resources

Interactive Element

Video: Converse, Inverse and Contrapositive of a Conditional Statement Principles - Basic

Activities: Converse, Inverse, and Contrapositive Discussion Questions

Study Aids: Conditional Statements Study Guide

Practice: Converse, Inverse, and Contrapositive Statements

Real World: Converse Inverse Contrapositive

Conditional Statements

Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?

Practice Problems & Videos

\(\textbf{1)}\) “if a figure has 3 sides, then it is a triangle.” state the hypothesis. show answer the hypothesis is “a figure has 3 sides.”, \(\textbf{2)}\) “if a figure has 3 sides, then it is a triangle.” state the conclusion. show answer the conclusion is “a figure is a triangle.”, \(\textbf{3)}\) “if a figure has 3 sides, then it is a triangle.” state the converse. show answer the converse is “if a figure is a triangle, then it has 3 sides.”, \(\textbf{4)}\) “if a figure has 3 sides, then it is a triangle.” state the inverse. show answer the inverse is “if a figure does not have 3 sides, then it is not a triangle.”, \(\textbf{5)}\) “if a figure has 3 sides, then it is a triangle.” state the contrapositive. show answer the contrapositive is “if a figure is not a triangle, then it does not have 3 sides.”, \(\textbf{6)}\) “if a figure has 3 sides, then it is a triangle.” state the biconditional. show answer the biconditional is “a figure has 3 sides, if and only if, it is a triangle.”, challenge problems, \(\textbf{7)}\) create a venn diagram for “all circles are ellipses.” show answer one example below, \(\textbf{8)}\) create a venn diagram for “if you don’t have an ellipse, then you don’t have a circle.” show answer note it is the same answer as number 7. they are equivalent statements., \(\textbf{9)}\) write 2 conditional statements based on the venn diagram below. show answer “if a square, then a rectangle.” or “all squares are rectangles” and “if not a rectangle, not a square.” or “all non-rectangles are non-squares”, see related pages\(\), \(\bullet\text{ geometry homepage}\) \(\,\,\,\,\,\,\,\,\text{all the best topics…}\), \(\bullet\text{ law of syllogism}\) \(\,\,\,\,\,\,\,\,\text{if p then q,}\) \(\,\,\,\,\,\,\,\,\text{if q then r,}\) \(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if p then r…}\), \(\bullet\text{ law of detachment}\) \(\,\,\,\,\,\,\,\,\text{if p then q,}\) \(\,\,\,\,\,\,\,\,\text{p is true,}\) \(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{q is true…}\), a conditional statement is a statement in the form “if p, then q,” where p and q are called the hypothesis and conclusion, respectively. the statement “if it is raining, then the ground is wet” is an example of a conditional statement. the converse of a conditional statement is formed by flipping the order in which the hypothesis and conclusion appear. for example, the converse of the statement “if it is raining, then the ground is wet” is “if the ground is wet, then it is raining.” the inverse of a conditional statement is formed by negating both the hypothesis and conclusion. for example, the inverse of the statement “if it is raining, then the ground is wet” is “if it is not raining, then the ground is not wet” the contrapositive of a conditional statement is formed by negating both the hypothesis and conclusion and flipping the order in which they appear. for example, the contrapositive of the statement “if it is raining, then the ground is wet” is “if the ground is not wet, then it is not raining.” a biconditional statement is a statement in the form “if and only if p, then q,” which is equivalent to the statement “p if and only if q.” this means that p and q are either both true or both false. for example, the statement “if and only if it is raining, the ground is wet” is a biconditional statement. in geometry class, students learn about conditional statements and their related concepts (inverse, converse, contrapositive, and biconditional) in order to make logical deductions about geometric figures and their properties. these concepts are often used to prove theorems and solve problems. andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. if you have any requests for additional content, please contact andy at [email protected] . he will promptly add the content. topics cover elementary math , middle school , algebra , geometry , algebra 2/pre-calculus/trig , calculus and probability/statistics . in the future, i hope to add physics and linear algebra content. visit me on youtube , tiktok , instagram and facebook . andymath content has a unique approach to presenting mathematics. the clear explanations, strong visuals mixed with dry humor regularly get millions of views. we are open to collaborations of all types, please contact andy at [email protected] for all enquiries. to offer financial support, visit my patreon page. let’s help students understand the math way of thinking thank you for visiting. how exciting.

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2.2: Logically Equivalent Statements

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Preview Activity \(\PageIndex{1}\): Logically Equivalent Statements

In Exercises (5) and (6) from Section 2.1, we observed situations where two different statements have the same truth tables. Basically, this means these statements are equivalent, and we make the following definition:

Two expressions are logically equivalent provided that they have the same truth value for all possible combinations of truth values for all variables appearing in the two expressions. In this case, we write \(X \equiv Y\) and say that \(X\) and \(Y\) are logically equivalent.

• Complete truth tables for \(\urcorner (P \wedge Q)\) and \(\urcorner P \vee \urcorner Q\).
• Are the expressions \(\urcorner (P \wedge Q)\) and \(\urcorner P \vee \urcorner Q\) logically equivalent?
• Suppose that the statement “I will play golf and I will mow the lawn” is false. Then its negation is true. Write the negation of this statement in the form of a disjunction. Does this make sense? Sometimes we actually use logical reasoning in our everyday living! Perhaps you can imagine a parent making the following two statements: Statement 1. If you do not clean your room, then you cannot watch TV. Statement 2. You clean your room or you cannot watch TV.
• Let \(P\) be “you do not clean your room,” and let \(Q\) be “you cannot watch TV.” Use these to translate Statement 1 and Statement 2 into symbolic forms.
• Construct a truth table for each of the expressions you determined in Part(4). Are the expressions logically equivalent?
• Assume that Statement 1 and Statement 2 are false. In this case, what is the truth value of \(P\) and what is the truth value of \(Q\)? Now, write a true statement in symbolic form that is a conjunction and involves \(P\) and \(Q\).
• Write a truth table for the (conjunction) statement in Part (6) and compare it to a truth table for \(\urcorner (P \to Q)\). What do you observe?

Preview Activity \(\PageIndex{2}\): Converse and Contrapositive

We now define two important conditional statements that are associated with a given conditional statement.

If \(P\) and \(Q\) are statements, then

• The converse of the conditional statement \(P \to Q\) is the conditional statement \(Q \to P\).
• The contrapositive of the conditional statement \(P \to Q\) is the conditional statement \(\urcorner Q \to \urcorner P\).
• For the following, the variable x represents a real number. Label each of the following statements as true or false. (a) If \(x = 3\), then \(x^2 = 9\). (b) If \(x^2 = 9\), then \(x = 3\). (c) If \(x^2 \ne 9\), then \(x \ne 3\). (d) If \(x \ne 3\), then \(x^2 \ne 9\).
• Which statement in the list of conditional statements in Part (1) is the converse of Statement (1a)? Which is the contrapositive of Statement (1a)?
• Complete appropriate truth tables to show that \(P \to Q\) is logically equivalent to its contrapositive \(\urcorner Q \to \urcorner P\). \(P \to Q\) is not logically equivalent to its converse \(Q \to P\)

In Preview Activity \(\PageIndex{1}\), we introduced the concept of logically equivalent expressions and the notation \(X \equiv Y\) to indicate that statements \(X\) and \(Y\) are logically equivalent. The following theorem gives two important logical equivalencies. They are sometimes referred to as De Morgan’s Laws .

Theorem 2.5: De Morgan’s Laws

For statements \(P\) and \(Q\),

• The statement \(\urcorner (P \wedge Q)\) is logically equivalent to \(\urcorner P \vee \urcorner Q\). This can be written as \(\urcorner (P \wedge Q) \equiv \urcorner P \vee \urcorner Q\).
• The statement \(\urcorner (P \vee Q)\) is logically equivalent to \(\urcorner P \wedge \urcorner Q\). This can be written as \(\urcorner (P \vee Q) \equiv \urcorner P \wedge \urcorner Q\).

The first equivalency in Theorem 2.5 was established in Preview Activity \(\PageIndex{1}\). Table 2.3 establishes the second equivalency.

It is possible to develop and state several different logical equivalencies at this time. However, we will restrict ourselves to what are considered to be some of the most important ones. Since many mathematical statements are written in the form of conditional statements, logical equivalencies related to conditional statements are quite important.

Logical Equivalencies Related to Conditional Statements

The first two logical equivalencies in the following theorem were established in Preview Activity \(\PageIndex{1}\), and the third logical equivalency was established in Preview Activity \(\PageIndex{2}\).

Theorem 2.6

• The conditional statement \(P \to Q\) is logically equivalent to \(\urcorner P \vee Q\).
• The statement \(\urcorner (P \to Q)\) is logically equivalent to \(P \wedge \urcorner Q\).
• The conditional statement \(P \to Q\) is logically equivalent to its contrapositive \(\urcorner Q \to \urcorner P\).

The Negation of a Conditional Statement

The logical equivalency \(\urcorner (P \to Q) \equiv P \wedge \urcorner Q\) is interesting because it shows us that the negation of a conditional statement is not another conditional statement . The negation of a conditional statement can be written in the form of a conjunction. So what does it mean to say that the conditional statement

If you do not clean your room, then you cannot watch TV,

is false? To answer this, we can use the logical equivalency \(\urcorner (P \to Q) \equiv P \wedge \urcorner Q\). The idea is that if \(P \to Q\) is false, then its negation must be true. So the negation of this can be written as

You do not clean your room and you can watch TV.

For another example, consider the following conditional statement:

If \(-5 < -3\), then \((-5)^2 < (-3)^2\).

This conditional statement is false since its hypothesis is true and its conclusion is false. Consequently, its negation must be true. Its negation is not a conditional statement. The negation can be written in the form of a conjunction by using the logical equivalency \(\urcorner (P \to Q) \equiv P \wedge \urcorner Q\). So, the negation can be written as follows:

\(5 < 3\) and \(\urcorner ((-5)^2 < (-3)^2)\).

However, the second part of this conjunction can be written in a simpler manner by noting that “not less than” means the same thing as “greater than or equal to.” So we use this to write the negation of the original conditional statement as follows:

\(5 < 3\) and \((-5)^2 \ge (-3)^2\).

This conjunction is true since each of the individual statements in the conjunction is true.

Another Method of Establishing Logical Equivalencies

We have seen that it often possible to use a truth table to establish a logical equivalency. However, it is also possible to prove a logical equivalency using a sequence of previously established logical equivalencies. For example,

• \(P \to Q\) is logically equivalent to \(\urcorner P \vee Q\). So
• \(\urcorner (P \to Q)\) is logically equivalent to \(\urcorner (\urcorner P \vee Q)\).
• Hence, by one of De Morgan’s Laws (Theorem 2.5), \(\urcorner (P \to Q)\) is logically equivalent to \(\urcorner (\urcorner P) \wedge \urcorner Q\).
• This means that \(\urcorner (P \to Q)\) is logically equivalent to\(P \wedge \urcorner Q\).

The last step used the fact that \(\urcorner (\urcorner P)\) is logically equivalent to \(P\).

When proving theorems in mathematics, it is often important to be able to decide if two expressions are logically equivalent. Sometimes when we are attempting to prove a theorem, we may be unsuccessful in developing a proof for the original statement of the theorem. However, in some cases, it is possible to prove an equivalent statement. Knowing that the statements are equivalent tells us that if we prove one, then we have also proven the other. In fact, once we know the truth value of a statement, then we know the truth value of any other logically equivalent statement. This is illustrated in Progress Check 2.7.

Progress Check 2.7 (Working with a logical equivalency)

In Section 2.1, we constructed a truth table for \((P \wedge \urcorner Q) \to R\).

• Although it is possible to use truth tables to show that \(P \to (Q \vee R)\) is logically equivalent to \(P \wedge \urcorner Q) \to R\), we instead use previously proven logical equivalencies to prove this logical equivalency. In this case, it may be easier to start working with \(P \wedge \urcorner Q) \to R\). Start with \(P \wedge \urcorner Q) \to R \equiv \urcorner (P \wedge \urcorner Q) \vee R\), which is justified by the logical equivalency established in Part (5) of Preview Activity 1. Continue by using one of De Morgan's Laws on \(\urcorner (P \wedge \urcorner Q)\).
• Let a and b be integers. Suppose we are trying to prove the following: If 3 is a factor of \(a \cdot b\), then 3 is a factor of \(a\) or 3 is a factor of \(b\). Explain why we will have proven this statement if we prove the following: If 3 is a factor of \(a \cdot b\) and 3 is not a factor of \(a\), then 3 is a factor of \(b\).

Add texts here. Do not delete this text first.

As we will see, it is often difficult to construct a direct proof for a conditional statement of the form \(P \to (Q \vee R)\). The logical equivalency in Progress Check 2.7 gives us another way to attempt to prove a statement of the form \(P \to (Q \vee R)\). The advantage of the equivalent form, \(P \wedge \urcorner Q) \to R\), is that we have an additional assumption, \(\urcorner Q\), in the hypothesis. This gives us more information with which to work.

Theorem 2.8 states some of the most frequently used logical equivalencies used when writing mathematical proofs.

Theorem 2.8: important logical equivalencies

For statement \(P\), \(Q\), and \(R\),

De Morgan's Laws \(\urcorner (P \wedge Q) \equiv \urcorner P \vee \urcorner Q\). \(\urcorner (P \vee Q) \equiv \urcorner P \wedge \urcorner Q\).

Conditional Statement. \(P \to Q \equiv \urcorner Q \to \urcorner P\) (contrapositive) \(P \to Q \equiv \urcorner P \vee Q\) \(\urcorner (P \to Q) \equiv P \wedge \urcorner Q\)

Biconditional Statement \((P leftrightarrow Q) \equiv (P \to Q) \wedge (Q \to P)\)

Double Negation \(\urcorner (\urcorner P) \equiv P\)

Distributive Laws \(P \vee (Q \wedge R) \equiv (P \vee Q) \wedge (P \vee R)\) \(P \wedge (Q \vee R) \equiv (P \wedge Q) \vee (P \wedge R)\)

Conditionals withDisjunctions \(P \to (Q \vee R) \equiv (P \wedge \urcorner Q) \to R\) \((P \vee Q) \to R \equiv (P \to R) \wedge (Q \to R)\)

We have already established many of these equivalencies. Others will be established in the exercises.

Exercises for Section 2.2

• Write the converse and contrapositive of each of the following conditional statements. (a) If \(a = 5\), then \(a^2 = 25\). (b) If it is not raining, then Laura is playing golf. (c) If \(a \ne b\), then \(a^4 \ne b^4\). (d) If \(a\) is an odd integer, then \(3a\) is an odd integer.
• Write each of the conditional statements in Exercise (1) as a logically equiva- lent disjunction, and write the negation of each of the conditional statements in Exercise (1) as a conjunction.
• Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.” (a) We will win the first game and we will win the second game. (b) They will lose the first game or they will lose the second game. (c) If you mow the lawn, then I will pay you $20. (d) If we do not win the first game, then we will not play a second game. (e) I will wash the car or I will mow the lawn. (f) If you graduate from college, then you will get a job or you will go to graduate school. (g) If I play tennis, then I will wash the car or I will do the dishes. (h) If you clean your room or do the dishes, then you can go to see a movie. (i) It is warm outside and if it does not rain, then I will play golf.
• Use truth tables to establish each of the following logical equivalencies dealing with biconditional statements: (a) \((P \leftrightarrow Q) \equiv (P \to Q) \wedge (Q \to P)\) (b) \((P \leftrightarrow Q) \equiv (Q \leftrightarrow P)\) (c) \((P \leftrightarrow Q) \equiv (\urcorner P \leftrightarrow \urcorner Q)\)
• Use truth tables to prove each of the distributive laws from Theorem 2.8. (a) \(P \vee (Q \wedge R) \equiv (P \vee Q) \wedge (P \vee R)\) (b \(P \wedge (Q \vee R) \equiv (P \wedge Q) \vee (P \wedge R)\)
• Use truth tables to prove the following logical equivalency from Theorem 2.8: \([(P \vee Q) \to R] \equiv (P \to R) \wedge (Q \to R)\).
• Use previously proven logical equivalencies to prove each of the following logical equivalencies about conditionals with conjunctions: (a) \([(P \wedge Q) \to R] \equiv (P \to R) \vee (Q \to R)\) (b) \([P \to (Q \wedge R)] \equiv (P \to R) \wedge (P \to R)\)
• If \(P\) and \(Q\) are statements, is the statement \((P \vee Q) \wedge \urcorner (P \wedge Q)\) logically equivalent to the statement \((P \wedge \urcorner Q) \vee (Q \wedge \urcorner P)\)? Justify your conclusion.
• Use previously proven logical equivalencies to prove each of the following logical equivalencies: (a) \([\urcorner P \to (Q \wedge \urcorner Q)] \equiv P\) (b) \((P \leftrightarrow Q) \equiv (\urcorner P \vee Q) \wedge (\urcorner Q \vee p)\) (c) \(\urcorner (P \leftrightarrow Q) \equiv (P \wedge \urcorner Q) \vee (Q \wedge \urcorner P)\) (d) \((P \to Q) \to R \equiv (P \wedge \urcorner Q) \vee R\) (e) \((P \to Q) \to R \equiv (\urcorner P \to R) \wedge (Q \to R)\) (f) \([(P \wedge Q) \to (R \vee S)] \equiv [(\urcorner R \wedge \urcorner S) \to (\urcorner P \vee \urcorner Q)]\) (g) \([(P \wedge Q) \to (R \vee S)] \equiv [(P \wedge Q \wedge \urcorner R) \to S]\) (h) \([(P \wedge Q) \to (R \vee S)] \equiv (\urcorner P \vee \urcorner Q \vee R \vee S)\) (i) \(\urcorner [(P \wedge Q) \to (R \vee S)] \equiv (P \wedge Q \wedge \urcorner R \wedge \urcorner S)\)

Which of the following statements have the same meaning as this conditional statement and which ones are negations of this conditional statement?

Note: This is not asking which statements are true and which are false. It is asking which statements are logically equivalent to the given statement. It might be helpful to let P represent the hypothesis of the given statement, \(Q\) represent the conclusion, and then determine a symbolic representation for each statement. Instead of using truth tables, try to use already established logical equivalencies to justify your conclusions. (a) If \(f\) is continuous at \(x = a\), then \(f\) is differentiable at \(x = a\). (b) If \(f\) is not differentiable at \(x = a\), then \(f\) is not continuous at \(x = a\). (c) If \(f\) is not continuous at \(x = a\), then \(f\) is not differentiable at \(x = a\). (d) \(f\) is not differentiable at \(x = a\) or \(f\) is continuous at \(x = a\). (e) \(f\) is not continuous at \(x = a\) or \(f\) is differentiable at \(x = a\). (f) \(f\) is differentiable at \(x = a\) or \(f\) is not continuous at \(x = a\).

The note for Exercise (10) also applies to this exercise. (a) If \(a\) divides \(b\) or \(a\) divides \(c\), then \(a\) divides \(bc\). (b) If \(a\) does not divide \(b\) or \(a\) does not divide \(c\), then \(a\) does not divide \(bc\). (c) \(a\) divides \(bc\), \(a\) does not divide \(b\), and \(a\) does not divide \(c\). (d) If \(a\) does not divide \(b\) and \(a\) does not divide \(c\), then \(a\) does not divide \(bc\). (e) \(a\) does not divide \(bc\) or \(a\) divides \(b\) or \(a\) divides \(c\). (f) If \(a\) divides \(bc\) and \(a\) does not divide \(c\), then \(a\) divides \(b\). (g) If \(a\) divides \(bc\) or \(a\) does not divide \(b\), then \(a\) divides \(c\).

• Let \(x\) be a real number. Consider the following conditional statement: If \(x^3 - x = 2x^2 +6\), then \(x = -2\) or \(x = 3\). Which of the following statements have the same meaning as this conditional statement and which ones are negations of this conditional statement? Explain each conclusion. (See the note in the instructions for Exercise (10).) (a) If \(x \ne -2\) and \(x \ne 3\), then \(x^3 - x \ne 2x^2 +6\). (b) If \(x = -2\) or \(x = 3\), then \(x^3 - x = 2x^2 +6\). (c) If \(x \ne -2\) or \(x \ne 3\), then \(x^3 - x \ne 2x^2 +6\). (d) If \(x^3 - x = 2x^2 +6\) and \(x \ne -2\), then \(x = 3\). (e) If \(x^3 - x = 2x^2 +6\) or \(x \ne -2\), then \(x = 3\). (f) \(x^3 - x = 2x^2 +6\), \(x \ne -2\), and \(x \ne 3\). (g) \(x^3 - x \ne 2x^2 +6\) or \(x = -2\) or \(x = 3\). Explorations and Activities

We notice that we can write this statement in the following symbolic form:

\(P \to (Q \vee R)\), where \(P\) is“\(x \cdot y\) is even,” \(Q\) is“\(x\) is even,”and \(R\) is “\(y\) is even.” (a) Write the symbolic form of the contrapositive of \(P \to (Q \vee R)\). Then use one of De Morgan’s Laws (Theorem 2.5) to rewrite the hypothesis of this conditional statement. (b) Use the result from Part (13a) to explain why the given statement is logically equivalent to the following statement: If \(x\) is odd and \(y\) is odd, then \(x \cdot y\) is odd. The two statements in this activity are logically equivalent. We now have the choice of proving either of these statements. If we prove one, we prove the other, or if we show one is false, the other is also false. The second statement is Theorem 1.8, which was proven in Section 1.2.

Contrapositive and Converse

You may come across different types of statements in mathematical reasoning where some are mathematically acceptable statements and some are not acceptable mathematically. In mathematics, we observe many statements with “if-then” frequently. For example, consider the statement. Contrapositive and converse are specific separate statements composed from a given statement with “if-then”. Before getting into the contrapositive and converse statements, let us recall what are conditional statements. A conditional statement is formed by “if-then” such that it contains two parts namely hypothesis and conclusion. Hypothesis exists in the”if” clause, whereas the conclusion exists in the “then” clause.

What are Contrapositive Statements?

It is easy to understand how to form a contrapositive statement when one knows about the inverse statement. To create the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. First, form the inverse statement, then interchange the hypothesis and the conclusion to write the conditional statement’s contrapositive.

Click here to know how to write the negation of a statement .

In other words, contrapositive statements can be obtained by adding “not” to both component statements and changing the order for the given conditional statements.

What are Converse Statements?

The converse statements are formed by interchanging the hypothesis and conclusion of given conditional statements.

Thus, we can relate the contrapositive, converse and inverse statements in such a way that the contrapositive is the inverse of a converse statement.

This can be better understood with the help of an example.

Example: Consider the following conditional statement.

If a number is a multiple of 8, then the number is a multiple of 4.

Write the contrapositive and converse of the statement.

Given conditional statement is:

The converse of the above statement is:

If a number is a multiple of 4, then the number is a multiple of 8.

The inverse of the given statement is obtained by taking the negation of components of the statement.

If a number is not a multiple of 8, then the number is not a multiple of 4.

Now, the contrapositive statement is:

If a number is not a multiple of 4, then the number is not a multiple of 8.

All these statements may or may not be true in all the cases. That means, any of these statements could be mathematically incorrect.

Contrapositive vs Converse

The differences between Contrapositive and Converse statements are tabulated below.

We can also construct a truth table for contrapositive and converse statement.

The truth table for Contrapositive of the conditional statement “If p, then q” is given below:

Similarly, the truth table for the converse of the conditional statement “If p, then q” is given as:

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Converse Statement

Converse Statement is a type of conditional statement where the hypothesis (or antecedent) and conclusion (or consequence) are reversed relative to a given conditional statement.

For instance, consider the statement: “If a triangle ABC is an equilateral triangle, then all its interior angles are equal.” The converse of this statement would be: “If all the interior angles of triangle ABC are equal, then it is an equilateral triangle”

In this article, we will discuss all the things related to the Converse statement in detail.

Table of Content

What is a Converse Statement?

How to write a converse statement, examples of converse statements.

• Truth Value of a Converse Statement

Truth Table for Converse Statement

• Other Types of Statements

A converse statement is a proposition formed by interchanging the hypothesis and conclusion of a conditional statement .

In simpler terms, it’s like flipping the order of “if” and “then” in a statement. For example, in the conditional statement “If it is raining, then the ground is wet”, the converse statement would be “If the ground is wet, then it is raining.”

Note: T he truth of the original statement doesn’t necessarily imply the truth of its converse, and vice versa.

Definition of Converse Statement

A converse statement is formed by exchanging the hypothesis and conclusion of a conditional statement while retaining the same meaning.

For instance, if the original statement is “If A, then B,” the converse is “If B, then A.” The validity of a converse statement doesn’t guarantee the truth of the original statement, and vice versa.

To write a converse statement, you simply switch the hypothesis and conclusion of a conditional statement while maintaining the same meaning. For example, if the original statement is “If it is raining (hypothesis), then the ground is wet (conclusion),” the converse statement would be “If the ground is wet (hypothesis), then it is raining (conclusion).” Remember, the converse statement may not always be true, even if the original statement is.

Some examples of converse statements are:

• Original Statement: If a shape is a square, then it has four equal sides. Converse Statement: If a shape has four equal sides, then it is a square.
• Original Statement: If it is summer, then the weather is hot. Converse Statement: If the weather is hot, then it is summer.
• Original Statement: If a number is divisible by 2, then it is even. Converse Statement: If a number is even, then it is divisible by 2.
• Original Statement: If a person is a teenager, then they are between 13 and 19 years old. Converse Statement: If a person is between 13 and 19 years old, then they are a teenager.
• Original Statement: If an animal is a dog, then it has fur. Converse Statement: If an animal has fur, then it is a dog.

Examples of Converse Statements in Mathematics or Logic

Some examples of converse statements in mathematics or logic:

• Original Statement: If two angles are congruent, then they have the same measure. Converse Statement: If two angles have the same measure, then they are congruent.
• Original Statement: If a number is divisible by 6, then it is divisible by 2 and 3. Converse Statement: If a number is divisible by 2 and 3, then it is divisible by 6.
• Original Statement: If two lines are perpendicular, then their slopes are negative reciprocals of each other. Converse Statement: If the slopes of two lines are negative reciprocals of each other, then the lines are perpendicular.

Converse, Inverse and Contrapositive Statements

Inverse Statement: The inverse of a conditional statement is formed by negating both the hypothesis and the conclusion of the original statement.

Contrapositive Statement: The contrapositive of a conditional statement is formed by switching the hypothesis and conclusion of the original statement and negating both.

Example of Inverse Statements

Original Statement: If a number is even, then it is divisible by 2. Inverse Statement : If a number is not even, then it is not divisible by 2.

Original Statement: If x > 5, then 2x > 10. Inverse Statement: If x ≤ 5, then 2x ≤ 10.

Example of Contrapositive Statements

Original Statement: If a shape is a square, then it has four equal sides. Contrapositive Statement: If a shape does not have four equal sides, then it is not a square.

Original Statement: If a number is even, then it is divisible by 2. Contrapositive: If a number is not divisible by 2, then it is not even.

To create a truth table for the converse statement, we need to consider both the original statement and its converse.

Let’s represent the original statement as “If p, then q” or “p → q” where p is the hypothesis and q is the conclusion. The converse of this statement is “If q, then p” or “q → p”. Then truth table is given by:

Truth Table for Inverse and Contrapositive Statement

To create a truth table for the inverse and contrapositive statements, let’s start with the original statement “If p, then q” or “p → q” where p is the hypothesis and q is the conclusion. The inverse of this statement is “If not p, then not q” or “~p → ~q”, and the contrapositive is “If not q, then not p” or “~q → ~p”. Then truth table is given by:

Solved Questions on Converse Statement

Example 1: If all squares are rectangles, are all rectangles squares?

Converse: If a shape is a rectangle, then it is a square.

The original statement says that all squares are rectangles. This is true because a square, by definition, has four sides of equal length and four right angles, making it a special type of rectangle where all sides are equal. However, the converse statement is not necessarily true. Not all rectangles are squares because rectangles can have unequal side lengths, whereas squares have all sides equal. Therefore, the converse statement is false.

Example 2: If all right angles are 90 degrees, are all 90 degree angles right angles?

Converse: If an angle measures 90 degrees, then it is a right angle.

The original statement is true because a right angle, by definition, measures 90 degrees. However, the converse statement is also true. If an angle measures 90 degrees, then it must be a right angle, as any angle measuring exactly 90 degrees forms a perfect right angle.

Example 3: If a number is divisible by 3, then it is an odd number.

Converse: If a number is an odd number, then it is divisible by 3.

The original statement is false. While it is true that all odd numbers are not divisible by 2, they are not necessarily divisible by 3. For example, the number 5 is an odd number but is not divisible by 3. Therefore, the converse statement is also false because not all odd numbers are divisible by 3.

Example 4: If a shape has four sides, then it is a quadrilateral.

Converse: If a shape is a quadrilateral, then it has four sides.

The original statement is true. A quadrilateral is defined as a polygon with four sides, so any shape with four sides is indeed a quadrilateral. Similarly, the converse statement is true. If a shape is a quadrilateral, then it must have four sides because that is a defining characteristic of a quadrilateral. Therefore, both the original statement and its converse are true.

Converse Statement: Practice Questions

Q1: If all birds have wings, do all winged creatures have beaks?

Q2: If all triangles have three sides, do all polygons with three sides have to be triangles?

Q3: If all vehicles are cars, are all cars vehicles?

Converse Statement: FAQs

What is conditional statement.

A conditional statement is a fundamental concept in logic and mathematics where a hypothesis is followed by a conclusion, often represented as “If p, then q.”

What is the Converse of a Statement?

The converse of a statement is formed by interchanging the antecedent and the consequent of a conditional statement. For example, if the original statement is “If it is raining, then the ground is wet,” the converse would be “If the ground is wet, then it is raining.”

How do Mathematicians Use Converse?

Mathematicians use the converse of a statement to explore the logical relationships between different assertions. By examining both the original statement and its converse, mathematicians can gain a deeper understanding of implications and relationships within a given context.

Is a Conditional Statement Logically Equivalent to a Converse and Inverse?

A conditional statement is not logically equivalent to its converse or inverse. While a conditional statement asserts a specific relationship between two events or conditions, the converse and inverse statements may or may not hold true in the same context.

Do the Converse and the Inverse Have The Same Truth Value?

The truth value of the converse and the inverse may differ from that of the original conditional statement. In some cases, the converse and the inverse of a true conditional statement may also be true, but this is not always the case. Each statement must be evaluated independently to determine its truth value.

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