hypothesis conclusion geometry

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

2.11: If Then Statements

• Last updated
• Save as PDF
• Page ID 2144

Hypothesis followed by a conclusion in a conditional statement.

Conditional Statements

A conditional statement (also called an if-then statement ) is a statement with a hypothesis followed by a conclusion . The hypothesis is the first, or “if,” part of a conditional statement. The conclusion is the second, or “then,” part of a conditional statement. The conclusion is the result of a hypothesis.

If-then statements might not always be written in the “if-then” form. Here are some examples of conditional statements:

• Statement 1: If you work overtime, then you’ll be paid time-and-a-half.
• Statement 2: I’ll wash the car if the weather is nice.
• Statement 3: If 2 divides evenly into \(x\), then \(x\) is an even number.
• Statement 4: I’ll be a millionaire when I win the lottery.
• Statement 5: All equiangular triangles are equilateral.

Statements 1 and 3 are written in the “if-then” form. The hypothesis of Statement 1 is “you work overtime.” The conclusion is “you’ll be paid time-and-a-half.” Statement 2 has the hypothesis after the conclusion. If the word “if” is in the middle of the statement, then the hypothesis is after it. The statement can be rewritten: If the weather is nice, then I will wash the car. Statement 4 uses the word “when” instead of “if” and is like Statement 2. It can be written: If I win the lottery, then I will be a millionaire. Statement 5 “if” and “then” are not there. It can be rewritten: If a triangle is equiangular, then it is equilateral.

What if you were given a statement like "All squares are rectangles"? How could you determine the hypothesis and conclusion of this statement?

Example \(\PageIndex{1}\)

Determine the hypothesis and conclusion: I'll bring an umbrella if it rains.

Hypothesis: "It rains." Conclusion: "I'll bring an umbrella."

Example \(\PageIndex{2}\)

Determine the hypothesis and conclusion: All right angles are \(90^{\circ}\).

Hypothesis: "An angle is right." Conclusion: "It is \(90^{\circ}\)."

Example \(\PageIndex{3}\)

Use the statement: I will graduate when I pass Calculus.

Rewrite in if-then form and determine the hypothesis and conclusion.

This statement can be rewritten as If I pass Calculus, then I will graduate. The hypothesis is “I pass Calculus,” and the conclusion is “I will graduate.”

Example \(\PageIndex{4}\)

Use the statement: All prime numbers are odd.

Rewrite in if-then form, determine the hypothesis and conclusion, and determine whether this is a true statement.

This statement can be rewritten as If a number is prime, then it is odd. The hypothesis is "a number is prime" and the conclusion is "it is odd". This is not a true statement (remember that not all conditional statements will be true!) since 2 is a prime number but it is not odd.

Example \(\PageIndex{5}\)

Determine the hypothesis and conclusion: Sarah will go to the store if Riley does the laundry.

The statement can be rewritten as "If Riley does the laundry then Sarah will go to the store." The hypothesis is "Riley does the laundry" and the conclusion is "Sarah will go to the store."

Determine the hypothesis and the conclusion for each statement.

• If 5 divides evenly into \(x\), then \(x\) ends in 0 or 5.
• If a triangle has three congruent sides, it is an equilateral triangle.
• Three points are coplanar if they all lie in the same plane.
• If \(x=3\), then \(x^2=9\).
• If you take yoga, then you are relaxed.
• All baseball players wear hats.
• I'll learn how to drive when I am 16 years old.
• If you do your homework, then you can watch TV.
• Alternate interior angles are congruent if lines are parallel.
• All kids like ice cream.

Additional Resources

Video: If-Then Statements Principles - Basic

Activities: If-Then Statements Discussion Questions

Study Aids: Conditional Statements Study Guide

Practice: If Then Statements

Real World: If Then Statements

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

Get access to all the courses and over 450 HD videos with your subscription

Monthly and Yearly Plans Available

Get My Subscription Now

Still wondering if CalcWorkshop is right for you? Take a Tour and find out how a membership can take the struggle out of learning math.

A free service from Mattecentrum

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"

• Angles, parallel lines and transversals
• Congruent triangles
• More about triangles
• Inequalities
• Mean and geometry
• The converse of the Pythagorean theorem and special triangles
• Properties of parallelograms
• Common types of transformation
• Transformation using matrices
• Basic information about circles
• Inscribed angles and polygons
• Advanced information about circles
• Parallelogram, triangles etc
• The surface area and the volume of pyramids, prisms, cylinders and cones
• SAT Overview
• ACT Overview

Rise and Sine

Conditional Statements in Geometry

Conditional statements in geometry can be confusing for even the best geometry students. The logic and proof portion of your geometry curriculum is bursting with new terminology! There are conditional statements, and the inverse, converse, contrapositive, etc. And wait, we represent them with p’s and q’s?! Ok, let’s break it down. 

What is a Conditional Statement?

A conditional statement in geometry is an “if-then” statement.

The part of the statement that follows “if” is called the hypothesis , and the part of the statement that follows “then” is called the conclusion .

We also represent conditional statements symbolically. For a conditional statement, p represents the hypothesis and q represents the conclusion. Symbolically we write p → q, which reads “if p then q.”

Statements Related to the Conditional Statement

  Inverse . To write the inverse of the conditional statement, you negate the hypothesis AND conclusion. Symbolically, it’s written as ~p → ~q and read as “If not p, then not q”.

Converse . To write the converse of the conditional statement, you switch the hypothesis and conclusion. Symbolically, it’s written as q → p and read “if q then p”.

Contrapositive . To write the contrapositive of the conditional statement, you both negate AND switch the hypothesis and conclusion. Symbolically, it’s written as ~q → ~p and read “if not q, then not p”. 

Resources for Teaching Conditional Statements

Looking for a graphic organizer to summarize conditional statements in geometry? Leave me your e-mail and I’ll send you one for FREE!  

Students can practice writing statements and determining their truth value with this self-checking assignment ! 

Stay tuned for a Logic and Proof Unit Bundle coming soon! 

Happy teaching!

Related posts

Self-Checking Math Resources: Why I Love Them and You Should Too!

Converting From Degrees to Radians (and How to Teach it!)

The Best Way to Teach Trig Identities to Your Precalc Class

No comments, leave a reply cancel reply.

Save my name, email, and website in this browser for the next time I comment.

The Role of Hypotheses and Conclusions in Mathematical Reasoning and Proof

Hypothesis and conclusion, in mathematics, a hypothesis is a statement or proposition that is assumed to be true and serves as the starting point for a reasoning or problem-solving process.

In mathematics, a hypothesis is a statement or proposition that is assumed to be true and serves as the starting point for a reasoning or problem-solving process. It is often denoted by the letter “H” or “P” and is used to build logical arguments or proofs. A hypothesis can be based on prior knowledge, observations, or assumptions, and it is subject to testing or investigation in order to determine its validity.

The conclusion, on the other hand, is the statement that follows logically from the hypothesis or premises. It is the result or outcome of the reasoning process and is typically denoted by the letter “C” or “Q.” The conclusion should be supported by evidence and be a logical consequence of the information provided.

When working with mathematical proofs, the hypothesis refers to the initial assumptions or statements you are given, and the conclusion is the statement that you aim to prove using logical deductions or mathematical techniques. In other words, the hypothesis sets the stage for the argument, while the conclusion is the final outcome or result.

For example, consider the following mathematical statement: “If a and b are even integers, then a + b is also an even integer.” Here, the hypothesis is “a and b are even integers,” and the conclusion is “a + b is an even integer.” To prove this statement, you would have to use logical reasoning or mathematical properties of even numbers.

In summary, a hypothesis is an assumption or starting point in mathematics, while the conclusion is the logical consequence or result that follows from the hypothesis. These concepts are fundamental in mathematical reasoning and are used to form logical arguments and proofs.

More Answers:

Recent posts, ramses ii a prominent pharaoh and legacy of ancient egypt.

Ramses II (c. 1279–1213 BCE) Ramses II, also known as Ramses the Great, was one of the most prominent and powerful pharaohs of ancient Egypt.

Formula for cyclic adenosine monophosphate & Its Significance

Is the formula of cyclic adenosine monophosphate (cAMP) $ce{C_{10}H_{11}N_{5}O_{6}P}$ or $ce{C_{10}H_{12}N_{5}O_{6}P}$? Does it matter? The correct formula for cyclic adenosine monophosphate (cAMP) is $ce{C_{10}H_{11}N_{5}O_{6}P}$. The

Development of a Turtle Inside its Egg

How does a turtle develop inside its egg? The development of a turtle inside its egg is a fascinating process that involves several stages and

The Essential Molecule in Photosynthesis for Energy and Biomass

Why does photosynthesis specifically produce glucose? Photosynthesis is the biological process by which plants, algae, and some bacteria convert sunlight, carbon dioxide (CO2), and water

How the Human Body Recycles its Energy Currency

Source for “The human body recycles its body weight of ATP each day”? The statement that “the human body recycles its body weight of ATP

Don't Miss Out! Sign Up Now!

Sign up now to get started for free!

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

2.6: Equivalent Statements

• Last updated
• Save as PDF
• Page ID 129512

Figure \(\PageIndex{1}\):  How your logical argument is stated affects the response, just like how you speak when holding a conversation can affect how your words are received. (credit: modification of work by Goelshivi/Flickr, Public Domain Mark 1.0)

Learning Objectives

After completing this section, you should be able to:

• Determine whether two statements are logically equivalent using a truth table.
• Compose the converse, inverse, and contrapositive of a conditional statement

Have you ever had a conversation with or sent a note to someone, only to have them misunderstand what you intended to convey? The way you choose to express your ideas can be as, or even more, important than what you are saying. If your goal is to convince someone that what you are saying is correct, you will not want to alienate them by choosing your words poorly.

Logical arguments can be stated in many different ways that still ultimately result in the same valid conclusion. Part of the art of constructing a persuasive argument is knowing how to arrange the facts and conclusion to elicit the desired response from the intended audience.

In this section, you will learn how to determine whether two statements are logically equivalent using truth tables, and then you will apply this knowledge to compose logically equivalent forms of the conditional statement. Developing this skill will provide the additional skills and knowledge needed to construct well-reasoned, persuasive arguments that can be customized to address specific audiences.

An alternate way to think about logical equivalence is that the truth values have to match. That is, whenever p p is true, q q is also true, and whenever p p is false, q q is also false.

Determine Logical Equivalence

Two statements, p p and q q , are logically equivalent when p ↔ q p ↔ q is a valid argument, or when the last column of the truth table consists of only true values. When a logical statement is always true, it is known as a tautology . To determine whether two statements p p and q q are logically equivalent, construct a truth table for p ↔ q p ↔ q and determine whether it valid. If the last column is all true, the argument is a tautology, it is valid, and p p is logically equivalent to q q ; otherwise, p p is not logically equivalent to q q .

Exercise \(\PageIndex{1}\)

Determining logical equivalence with a truth table.

Create a truth table to determine whether the following compound statements are logically equivalent.

• p → q ; p → q ; ~ p → ~ q ~ p → ~ q
• p → q ; p → q ; ~ p ∨ q

1. Construct a truth table for the biconditional formed by using the first statement as the hypothesis and the second statement as the conclusion, ( p → q ) ↔ ( ~ p → ~ q ).

Because the last column it not all true, the biconditional is not valid and the statement p → q p → q is not logically equivalent to the statement ~ p → ~ q ~ p → ~ q .

2. Construct a truth table for the biconditional formed by using the first statement as the hypothesis and the second statement as the conclusion, ( p → q) ↔ ( ~ p ∨ q ) . 

Because the last column is true for every entry, the biconditional is valid and the statement p → q p → q is logically equivalent to the statement ~ p ∨ q ~ p ∨ q . Symbolically, p → q ≡ ~ p ∨ q .

Your Turn \(\PageIndex{1}\)

1. \(p \rightarrow q ; q \rightarrow \sim p\) 2. \(p \rightarrow q ; p \vee \sim q\)

Compose the Converse, Inverse, and Contrapositive of a Conditional Statement

The converse , inverse , and contrapositive are variations of the conditional statement, p → q . p → q .

• The converse is if q q then p p , and it is formed by interchanging the hypothesis and the conclusion. The converse is logically equivalent to the inverse.
• The inverse is if ~ p ~ p then ~ q ~ q , and it is formed by negating both the hypothesis and the conclusion. The inverse is logically equivalent to the converse.
• The contrapositive is if ~ q ~ q then ~ p ~ p , and it is formed by interchanging and negating both the hypothesis and the conclusion. The contrapositive is logically equivalent to the conditional.

The table below shows how these variations are presented symbolically.

Exercise \(\PageIndex{2}\)

Writing the converse, inverse, and contrapositive of a conditional statement.

Use the statements, p p : Harry is a wizard and q q : Hermione is a witch, to write the following statements:

• Write the conditional statement, p → q p → q , in words.
• Write the converse statement, q → p q → p , in words.
• Write the inverse statement, ~ p → ~ q ~ p → ~ q , in words.
• Write the contrapositive statement, ~ q → ~ p ~ q → ~ p , in words.
• The conditional statement takes the form, “if p p , then q q ,” so the conditional statement is: “If Harry is a wizard, then Hermione is a witch.” Remember the if … then … words are the connectives that form the conditional statement.
• The converse swaps or interchanges the hypothesis, p p , with the conclusion, q q . It has the form, “if q q , then p p .” So, the converse is: "If Hermione is a witch, then Harry is a wizard."
• To construct the inverse of a statement, negate both the hypothesis and the conclusion. The inverse has the form, “if ~ p ~ p , then ~ q ~ q ,” so the inverse is: "If Harry is not a wizard, then Hermione is not a witch."
• The contrapositive is formed by negating and interchanging both the hypothesis and conclusion. It has the form, “if ~ q ~ q , then ~ p ~ p ," so the contrapositive statement is: "If Hermione is not a witch, then Harry is not a wizard."

Your Turn \(\PageIndex{2}\)

Use the statements, \(p\) : Elvis Presley wore capes and \(q\) : Some superheroes wear capes, to write the following statements:

1. Write the conditional statement, \(p \rightarrow q\), in words. 2. Write the converse statement, \(q \rightarrow p\), in words. 3. Write the inverse statement, \(\sim p \rightarrow \sim q\), in words. 4. Write the contrapositive statement, \(\sim q \rightarrow \sim p\), in words.

Exercise \(\PageIndex{3}\)

Use the conditional statement, “If all dogs bark, then Lassie likes to bark,” to identify the following.

• Write the hypothesis of the conditional statement and label it with a p p .
• Write the conclusion of the conditional statement and label it with a q q .
• Identify the following statement as the converse, inverse, or contrapositive: “If Lassie likes to bark, then all dogs bark.”
• Identify the following statement as the converse, inverse, or contrapositive: “If Lassie does not like to bark, then some dogs do not bark.”
• Which statement is logically equivalent to the conditional statement?
• The hypothesis is the phrase following the if . The answer is p : All dogs bark. Notice, the word if is not included as part of the hypothesis.
• The conclusion of a conditional statement is the phrase following the then . The word then is not included when stating the conclusion. The answer is: q: Lassie likes to bark.
• “Lassie likes to bark” is q  and “All dogs bark” is p . So, “If Lassie likes to bark, then all dogs bark,” has the form “if q , then p ,” which is the form of the converse. “Lassie does not like to bark” is ~ q  and “Some dogs do not bark” is ~ p . The statement, “If Lassie does not like to bark, then some dogs do not bark,” has the form “if ~ q , then ~ p ,” which is the form of the contrapositive.
• The contrapositive ~ q → ~ p is logically equivalent to the conditional statement p → q .

Your Turn \(\PageIndex{3}\)

Use the conditional statement, “If Dora is an explorer, then Boots is a monkey,” to identify the following:

1 . Write the hypothesis of the conditional statement and label it with a p�.

2 . Write the conclusion of the conditional statement and label it with a q�.

3 . Identify the following statement as the converse, inverse, or contrapositive: “If Dora is not an explorer, then Boots is not a monkey.”

4 . Identify the following statement as the converse, inverse, or contrapositive: “If Boots is a monkey, then Dora is an explorer.”

5 . Which statement is logically equivalent to the inverse?

Exercise \(\PageIndex{4}\)

Assume the conditional statement, p→q:: “If Chadwick Boseman was an actor, then Chadwick Boseman did not star in the movie  Black Panther ” is false, and use it to answer the following questions.

• Write the converse of the statement in words and determine its truth value.
• Write the inverse of the statement in words and determine its truth value.
• Write the contrapositive of the statement in words and determine its truth value.

The only way the conditional statement can be false is if the hypothesis, p : Chadwick Boseman was an actor, is true and the conclusion, q: Chadwick Boseman did not star in the movie Black Panther , is false. The converse is q → p, and it is written in words as: “If Chadwick Boseman did not star in the movie Black Panther , then Chadwick Boseman was an actor.” This statement is true, because false → true is true.

The inverse has the form “ ~ p → ~ q. ” The written form is: “If Chadwick Boseman was not an actor, then Chadwick Boseman starred in the movie Black Panther .” Because p is true, and q is false, ~ p  is false, and ~ q  is true. This means the inverse is false →  true, which is true. Alternatively, from Question 1, the converse is true, and because the inverse is logically equivalent to the converse it must also be true.

The contrapositive is logically equivalent to the conditional. Because the conditional is false, the contrapositive is also false. This can be confirmed by looking at the truth values of the contrapositive statement. The contrapositive has the form “ ~ q → ~ p. ” Because q is false and p is true, ~ q is true and ~ p  is false. Therefore, ~ q → ~ p  is true →  false, which is false. The written form of the contrapositive is “If Chadwick Boseman starred in the movie Black Panther , then Chadwick Boseman was not an actor.”

Your Turn \(\PageIndex{4}\)

Assume the conditional statement  p→q : “If my friend lives in San Francisco, then my friend does not live in California” is false, and use it to answer the following questions.

1 . Write the converse of the statement in words and determine its truth value.

2 . Write the inverse of the statement in words and determine its truth value.

3 . Write the contrapositive of the statement in words and determine its truth value.

Check Your Understanding

• Two statements p and q are logically equivalent to each other if the biconditional statement, p \leftrightarrow q is ________________.   
• The _____ statement has the form, “ p then q .”  
• The contrapositive is _____________ ___________ to the conditional statement, and has the form, "if ~q , then ~p ."  
• The _________________ of the conditional statement has the form, "if ~p , then ~q ."  
• The _________________ of the conditional statement is logically equivalent to the _______________ and has the form, "if q then p ."

Section 2.5 Exercises

For the following exercises, determine whether the pair of compound statements are logically equivalent by constructing a truth table.

Professor: Erika L.C. King Email: [email protected] Office: Lansing 304 Phone: (315)781-3355

The majority of statements in mathematics can be written in the form: "If A, then B." For example: "If a function is differentiable, then it is continuous". In this example, the "A" part is "a function is differentiable" and the "B" part is "a function is continuous." The "A" part of the statement is called the "hypothesis", and the "B" part of the statement is called the "conclusion". Thus the hypothesis is what we must assume in order to be positive that the conclusion will hold.

Whenever you are asked to state a theorem, be sure to include the hypothesis. In order to know when you may apply the theorem, you need to know what constraints you have. So in the example above, if we know that a function is differentiable, we may assume that it is continuous. However, if we do not know that a function is differentiable, continuity may not hold. Some theorems have MANY hypotheses, some of which are written in sentences before the ultimate "if, then" statement. For example, there might be a sentence that says: "Assume n is even." which is then followed by an if,then statement. Include all hypotheses and assumptions when asked to state theorems and definitions!

Still have questions? Please ask!

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.

RELATED POSTS

• Ordering Decimals: Definition, Types, Examples
• Decimal to Octal: Steps, Methods, Conversion Table
• Lattice Multiplication – Definition, Method, Examples, Facts, FAQs
• X Intercept – Definition, Formula, Graph, Examples
• Lateral Face – Definition With Examples

Math & ELA | PreK To Grade 5

Kids see fun., you see real learning outcomes..

Make study-time fun with 14,000+ games & activities, 450+ lesson plans, and more—free forever.

Parents, Try for Free Teachers, Use for Free

Biconditional Statement — Definition, Examples & How To Write

What is a biconditional statement?

A biconditional statement combines a conditional statement with its converse statement. Both the conditional and converse statements must be true to produce a biconditional statement.

If we remove the  if-then  part of a true conditional statement, combine the hypothesis and conclusion, and tuck in a phrase "if and only if," we can create  biconditional statements .

Geometry and logic cross paths many ways. One example is a  biconditional statement . To understand biconditional statements, we first need to review conditional and converse statements. Then we will see how these logic tools apply to geometry.

Conditional statements

In logic, concepts can be conditional, using an  if-then  statement:

If I have a pet goat, then my homework will be eaten.

If I have a triangle, then my polygon has only three sides.

If the polygon has only four sides, then the polygon is a quadrilateral.

If I eat lunch, then my mood will improve.

If I ask more questions in class, then I will understand the mathematics better.

If the quadrilateral has four congruent sides and angles, then the quadrilateral is a square.

Each of these conditional statements has a  hypothesis  ("If …") and a  conclusion  (" …, then …").

These statements can be true or false. Whether the conditional statement is true or false does not matter (well, it will eventually), so long as the second part (the conclusion) relates to, and is dependent on, the first part (the hypothesis).

Converse statements

To create a  converse statement  for a given conditional statement, switch the hypothesis and the conclusion. You may "clean up" the two parts for grammar without affecting the logic.

Take the first conditional statement from above:

Hypothesis: If I have a pet goat …

Conclusion: … then my homework will be eaten.

Create the converse statement:

Hypothesis: If my homework is eaten …

Conclusion: Then I have a pet goat.

Converse: If my homework is eaten, then I have a pet goat.

This converse statement is not true, as you can conceive of something … or someone … else eating your homework: your dog, your little brother. Your homework being eaten does not  automatically  mean you have a goat.

Let's apply the same concept of switching conclusion and hypothesis to one of the conditional geometry statements:

Conditional: If I have a triangle, then my polygon has only three sides.

Converse: If my polygon has only three sides, then I have a triangle.

This converse is true; remember, though, neither the original conditional statement nor its converse have to be true to be valid, logical statements.

Converse statement examples

For, "If the polygon has only four sides, then the polygon is a quadrilateral," write the converse statement.

Converse:  If the polygon is a quadrilateral, then the polygon has only four sides.

Try this one, too: "If the quadrilateral has four congruent sides and angles, then the quadrilateral is a square."

Converse:  If the quadrilateral is a square, then the quadrilateral has four congruent sides and angles.

How to write a biconditional statement

The general form (for goats, geometry or lunch) is:

Hypothesis  if and only if  conclusion .

Because the statement is biconditional (conditional in both directions), we can also write it this way, which is the converse statement:

Conclusion  if and only if  hypothesis .

Notice we can create  two  biconditional statements. If conditional statements are one-way streets, biconditional statements are the two-way streets of logic.

Both the conditional and converse statements must be true to produce a biconditional statement.

Conditional: If I have a triangle, then my polygon has only three sides. (true)

Converse: If my polygon has only three sides, then I have a triangle. (true)

Since both statements are true, we can write two biconditional statements:

I have a triangle if and only if my polygon has only three sides. (true)

My polygon has only three sides if and only if I have a triangle. (true)

You can do this if and only if both conditional and converse statements have the same truth value. They could both be  false  and you could still write a true biconditional statement ("My pet goat draws polygons if and only if my pet goat buys art supplies online.").

Let's see how different truth values prevent logical biconditional statements, using our pet goat:

Conditional: If I have a pet goat, then my homework will be eaten. (true)

Converse: If my homework is eaten, then I have a pet goat. (not true)

We can attempt, but fail to write, logical biconditional statements, but they will not make sense:

I have a pet goat if and only if my homework is eaten. (not true)

My homework will be eaten if and only if I have a pet goat. (not true)

Biconditional statement symbols

You may recall that logic symbols can replace words in statements. So the conditional statement, "If I have a pet goat, then my homework gets eaten" can be replaced with a  p  for the hypothesis, a  q  for the conclusion, and a  → \to →  for the connector:

For biconditional statements, we use a double arrow, ⇔ \Leftrightarrow ⇔ , since the truth works in both directions:

Biconditional statement examples

We still have several conditional geometry statements and their converses from above.

Conditional: If the polygon has only four sides, then the polygon is a quadrilateral. (true)

Converse: If the polygon is a quadrilateral, then the polygon has only four sides. (true)

Conditional: If the quadrilateral has four congruent sides and angles, then the quadrilateral is a square. (true)

Converse: If the quadrilateral is a square, then the quadrilateral has four congruent sides and angles. (true)

Try your hand at these first, then check below. The biconditional statements for these two sets would be:

The polygon has only four sides if and only if the polygon is a quadrilateral.

The polygon is a quadrilateral if and only if the polygon has only four sides.

The quadrilateral has four congruent sides and angles if and only if the quadrilateral is a square.

The quadrilateral is a square if and only if the quadrilateral has four congruent sides and angles.

More examples

See if you can write the converse and biconditional statements for these. You can "clean up" the words for grammar.

Try doing it before peeking below!

If I eat lunch, then my mood will improve. (true)

If my mood improves, then I will eat lunch. (true)

Biconditional statements:

I will eat lunch if and only if my mood improves.

My mood will improve if and only if I eat lunch.

And now the other leftover:

If I ask more questions in class, then I will understand the mathematics better. (true)

If I understand the mathematics better, then I will ask more questions in class. (false)

You cannot write a biconditional statement for this leftover; the truth values are not the same.

• Mathematicians
• Math Lessons
• Square Roots
• Math Calculators
• Conclusion | Definition & Meaning

JUMP TO TOPIC

Hypothesis and Conclusion

If-then statement, a implies b, conclusion|definition & meaning.

 The term conclusion in maths is used to define us about the problem that we solve and when we produce the final result at the end then that stage of processes is called as conclusion.

Figure 1 – Give the Right Conclusion to the problem 

When you solve a maths question, you have to end the problem by calculating the last answer and pulling a conclus ion by writing the answer.  A conclusion is the last step of the maths problem. The conclusion is the final answer produced in the end . The answer is completed by writing the arguments and statements by telling the answer to the question. The ending statement of a problem is called a conclusion.

Drawing conclusions refers to the act of thinking of interpreting a series of premises or some ideas and, from them, suggesting something that leads to a meaningful finding. It is normally regarded as a conscious way of learning .

As a rule, a mathematical statement comprises two sections : the first section is assumptions or hypotheses , and the other section is the conclusion . Most mathematical statements have the form “If A, then B.” Often, this statement is written as “A implies B” or “A $\Rightarrow $ B.”  The assumptions we make are what makes “A,” and the circumstances that make “B” are called the conclusion .

To prove that a given statement “If A, then B” is said to be true, we will require some assumptions for “A,” and after doing some work on it, we need to conclude that “B” must also hold when “A” holds.

If we are asked to apply the statement “If A, then B,” firstly, we should be sure that the conditions of the statement “A” are met and true before we start to talk about the conclusion “B.”

Suppose you want to apply the statement “x is even $\Rightarrow$ x2 is an integer.” First, you must verify  that x is even  before  you  conclude that x2 is an integer.

In maths, you will, at many times, confront statements in the form “X $\Leftrightarrow$ Y” or “X if and only if Y.”  These statements are actually two “if, then” statements. The following statement, “X if and only if Y,” is logically equivalent to the statements “If X, then Y” and “If Y, then X.” One more method for thinking about this kind of explanation is an equality between the statements X and Y: so, whenever X holds, Y holds, and whenever Y hold, X holds.

Assume the example: “ x is even $\Leftrightarrow$ x 2  is an integer “. Statement A says, “ x is even,” whereas statement B says, “ x 2  is an integer.” If we get a quick revision about what it suggests to be even (simply that x is a multiple of 2), we can see with ease that the following two statements are identical : If x = 2 k is proved to be even, then it implies x 2 = 2 k 2 = k is an integer, and we know that x 2 = k is an integer, then x = 2 k so n is proved to be even.

In day-to-day use, a statement which is in the form “ If A, then B ,” in some cases, means “ A if and only if B. ” For example, when people agree on a deal, they say, “If you agree to sell me your car for 500k, then I’ll buy from you this week” they straightaway mean, “I’ll buy your car if and only if you agree to sell me in 500k.” In other words, if you don’t agree on 500k, they will not be buying your car from you .

In geometry, the validation or proof is stated in the if-then format. The “if” is a condition or hypothesis , and if that condition is met, only then the second part of the statement is true , which is called the conclusion . The working is like any other if-then statement. For illustration, the statement “If a toy shop has toys for two age groups and 45 percent of toys in the shop are for 14 or above years old, then 55 percent of the toys in the shop are for 13 and fewer years old.” The above statement concludes that “55 percent of the toys in the shop are for 13 and fewer years old.”

In maths, the statement “A if and only if B” is very different from “A implies B.” Assume the example: “ x is an integer” is the A statement, and “ x 3 is a rational number” is the B statement  The statement “A implies B” here means “If x is an integer, then x 3 is a rational number.” The statement is proven to be true. On the other hand, the statement, “A, if and only if B,” means “ x is an integer if and only if x 3 is a rational number,” which is not true in this case.

Examples of Drawing Conclusions

Consider the equation below. Comment if this equation is true or false.

Figure 3 – Example Problem

To calculate its true answer, first, consider the hypothesis $x>0$. Whatever we are going to conclude, it will be a consequence of the truth that $x$ is positive.

Next, consider the conclusion $x+1>0$. This equation is right, since $x+1>x>0$.

This implies that the provided inequality is true.

Simplify the below problem by providing a conclusion by calculating the answer of A.

\[ A= \dfrac{35}{3} \]

The expression given in the question is: $A= \dfrac{35}{3}$

Calculating the answer of A to make a conclusion, The arithmetic operation division is found in the question that is to be figured out in the provided problem. After figuring out the answer to expression A, The conclusion will be given.

\[ A= 11. 667 \]

Therefore, we conclude the question by calculating the answer of $A=11.666$

Consider the equation $0>1 \Rightarrow sinx=2$. Is this equation true or false?

To calculate the correct answer, first consider the hypothesis $0>1$. This equation is clearly false.

calculate the below problem by providing a conclusion by estimating the value of X.

\[ 3+8 \times 2\]

The expression given in the problem is $3+8 \times 2 $.

Multiplication and Plus operation is to be carried out to calculate the answer to the given problem. After figuring out the answer to X  the conclusion will be given.

Thus, we conclude the example by calculating the value of $X = 19$.

All images/mathematical drawings were created with GeoGebra.

Concentric Circles Definition < Glossary Index > Cone Definiton