hypothesis and conclusion of the following 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.

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

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

Step-by-step Guide: Conditional Statements

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

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

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

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

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

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

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

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

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

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

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

Practice Questions:

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

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

Effortless Math Team

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Algebra I Study Guide A Comprehensive Review and Step-By-Step Guide to Preparing for Algebra I

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3.2: More Methods of Proof

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• Page ID 7047

• Ted Sundstrom
• Grand Valley State University via ScholarWorks @Grand Valley State University

Preview Activity 1: Using the Contrapositive

The following statement was proven in Exercise (3c) on page 27 in Section 1.2.

If \(n\) is an odd integer, then \(n^2\) is an odd integer.

Now consider the following proposition:

For each integer \(n\), if \(n^2\) is an odd integer, then \(n\) is an odd integer.

• After examining several examples, decide whether you think this proposition is true or false.
• Try completing the following know-show table for a direct proof of this proposition. The question is, “Can we perform algebraic manipulations to get from the ‘know’ portion of the table to the ‘show’ portion of the table?” Be careful with this! Remember that we are working with integers and we want to make sure that we can end up with an integer q as stated in Step \(Q\)1.

Recall that the contrapositive of the conditional statement \(P \to Q\) is the conditional statement \(\urcorner Q \to \urcorner P\). We have seen in Section 2.2 that the contrapositive of a conditional statement is logically equivalent to the conditional statement. (It might be a good idea to review Preview Activity \(\PageIndex{2}\) from Section 2.2 on page 44.) Consider the following proposition once again: For each integer \(n\), if \(n^2\) is an odd integer, then \(n\) is an odd integer.

• Write the contrapositive of this conditional statement. Remember that “not odd” means “even.”
• Complete a know-show table for the contrapositive statement from Part(3).
• By completing the proof in Part (4), have you proven the given proposition? That is, have you proven that if \(n^2\) is an odd integer, then \(n\) is an odd integer? Explain.

Preview Activity 2: A Biconditional Statement

• In Exercise (4a) from Section 2.2, we constructed a truth table to prove that the biconditional statement, \(P \leftrightarrow Q\), is logically equivalent to \(P \to Q) \wedge (Q \to P\). Complete this exercise if you have not already done so.
• Suppose that we want to prove a biconditional statement of the form \(P \leftrightarrow Q\). Explain a method for completing this proof based on the logical equivalency in part (1).
• If n is an odd integer, then n2 is an odd integer.
• If n2 is an odd integer, then n is an odd integer.

(See Exercise (3c) from Section 1.2 and Preview Activity \(\PageIndex{1}\).) Have we completed the proof of the following proposition?

For each integer \(n\), \(n\) is an odd integer if and only if \(n^2\) is an odd integer. Explain.

Review of Direct Proofs

In Sections 1.2 and 3.1, we studied direct proofs of mathematical statements. Most of the statements we prove in mathematics are conditional statements that can be written in the form \(P \to Q\). A direct proof of a statement of the form \(P \to Q\) is based on the definition that a conditional statement can only be false when the hypothesis, \(P\), is true and the conclusion, \(Q\), is false. Thus, if the conclusion is true whenever the hypothesis is true, then the conditional statement must be true. So, in a direct proof,

• We start by assuming that \(P\) is true.
• From this assumption, we logically deduce that \(Q\) is true.

We have used the so-called forward and backward method to discover how to logically deduce \(Q\) from the assumption that \(P\) is true.

Proof Using the Contrapositive

As we saw in Preview Activity \(\PageIndex{1}\), it is sometimes difficult to construct a direct proof of a conditional statement. This is one reason we studied logical equivalencies in Section 2.2. Knowing that two expressions are logically 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 statement that is logically equivalent to it.

One of the most useful logical equivalencies in this regard is that a conditional statement \(P \to Q\) is logically equivalent to its contrapositive, \(\urcorner Q \to \urcorner P\). This means that if we prove the contrapositive of the conditional statement, then we have proven the conditional statement. The following are some important points to remember.

• A conditional statement is logically equivalent to its contrapositive.
• Use a direct proof to prove that \(\urcorner Q \to \urcorner P\) is true.
• Caution: One difficulty with this type of proof is in the formation of correct negations. (We need to be very careful doing this.)
• We might consider using a proof by contrapositive when the statements \(P\) and \(Q\) are stated as negations.

Writing Guidelines

One of the basic rules of writing mathematical proofs is to keep the reader informed. So when we prove a result using the contrapositive, we indicate this within the first few lines of the proof. For example,

• We will prove this theorem by proving its contrapositive.
• We will prove the contrapositive of this statement.

In addition, make sure the reader knows the status of every assertion that you make. That is, make sure you state whether an assertion is an assumption of the theorem, a previously proven result, a well-known result, or something from the reader’s mathematical background. Following is a completed proof of a statement from Preview Activity \(\PageIndex{1}\).

Theorem 3.7

For each integer \(n\), if \(n^2\) is an even integer, then \(n\) is an even integer.

We will prove this result by proving the contrapositive of the statement, which is

For each integer \(n\), if \(n\) is an odd integer, then \(n^2\) is an odd integer.

However, in Theorem 1.8 on page 21, we have already proven that if \(x\) and \(y\) are odd integers, then \(x \cdot y\) is an odd integer. So using \(x = y = n\), we can conclude that if \(n\) is an odd integer, then \(n \cdot n\), or \(n^2\), is an odd integer. We have thus proved the contrapositive of the theorem, and consequently, we have proved that if \(n^2\) is an even integer, then \(n\) is an even integer.

Using Other Logical Equivalencies

As was noted in Section 2.2, there are several different logical equivalencies. Fortunately, there are only a small number that we often use when trying to write proofs, and many of these are listed in Theorem 2.8 at the end of Section 2.2. We will illustrate the use of one of these logical equivalencies with the following proposition:

For all real numbers \(a\) and \(b\), if \(a \ne 0\) and \(b \ne 0\), then \(ab \ne 0\).

First, notice that the hypothesis and the conclusion of the conditional statement are stated in the form of negations. This suggests that we consider the contrapositive. Care must be taken when we negate the hypothesis since it is a conjunction. We use one of De Morgan’s Laws as follows:

\[\urcorner (a \ne 0 \wedge b \ne 0) \equiv (a = 0) \vee (b = 0).\]

Progress Check 3.8 (Using Another Logical Equivalency)

The contrapositive is a conditional statement in the form \(X \to (Y \vee Z\). The difficulty is that there is not much we can do with the hypothesis \(ab = 0\) since we know nothing else about the real numbers \(a\) and \(b\). However, if we knew that \(a\) was not equal to zero, then we could multiply both sides of the equation \(ab = 0\) by \(\dfrac{1}{a}\). This suggests that we consider using the following logical equivalency based on a result in Theorem 2.8 on page 48: \[X \to (Y \vee Z) \equiv (X \wedge \urcorner Y) \to Z.\]

The logical equivalency in Part (2) makes sense because if we are trying to prove \(Y \vee Z\), we only need to prove that at least one of \(Y\) or \(Z\) is true. So the idea is to prove that if \(Y\) is false, then \(Z\) must be true.

Proposition 3.9.

For all real numbers \(a\) and \(b\), if \(a \ne\) and \(b \ne 0\), then \(ab \ne 0\).

We will prove the contrapositive of this proposition, which is

For all real numbers \(a\) and \(b\), if \(ab = 0\), then \(a = 0\) or \(b = 0\).

This contrapositive, however, is logically equivalent to the following:

For all real numbers \(a\) and \(b\), if \(ab = 0\) and \(a \ne 0\), then \(b = 0\).

To prove this, we let \(a\) and \(b\) be real numbers and assume that \(ab = 0\) and\(a \ne 0\). We can then multiply both sides of the equation \(ab = 0\) by \(\dfrac{1}{a}\). This gives

Now complete the proof.

Therefore, \(b = 0\). This completes the proof of a statement that is logically equivalent to the contrapositive, and hence, we have proven the proposition.

Add texts here. Do not delete this text first.

Proofs of Biconditional Statements

In Preview Activity \(\PageIndex{2}\), we used the following logical equivalency:

\[(P \leftrightarrow Q) \equiv (P \to Q) \wedge (Q \to P).\]

This logical equivalency suggests one method for proving a biconditional statement written in the form “\(P\) if and only if \(Q\).” This method is to construct separate proofs of the two conditional statements \(P \to Q\) and \(Q \to P\). For example, since we have now proven each of the following:

• For each integer \(n\), if \(n\) is an even integer, then \(n^2\) is an even integer. (Exercise (3c) on page 27 in Section 1.2)
• For each integer \(n\), if \(n^2\) is an even integer, then \(n\) is an even integer. (Theorem 3.7)

We can state the following theorem.

Theorem 3.10.

For each integer \(n\), \(n\) is an even integer if and only if \(n^2\) is an even integer.

When proving a biconditional statement using the logical equivalency \((P \leftrightarrow Q) \equiv (P \to Q) \wedge (Q \to P)\), we actually need to prove two conditional statements. The proof of each conditional statement can be considered as one of two parts of the proof of the biconditional statement. Make sure that the start and end of each of these parts is indicated clearly. This is illustrated in the proof of the following proposition.

Proposition 3.11

Let \(x \in \mathbb{R}\). The real number \(x\) equals 2 if and only if \(x^3 - 2x^2 + x = 2\).

We will prove this biconditional statement by proving the following two conditional statements:

• For each real number \(x\), if \(x\) equals 2, then \(x^3 - 2x^2 + x = 2\).
• For each real number \(x\), if \(x^3 - 2x^2 + x = 2\), then \(x\) equals 2.

For the first part, we assume \(x = 2\) and prove that \(x^3 - 2x^2 + x = 2\). We can do this by substituting \(x = 2\) into the expression \(x^3 - 2x^2 + x\). This gives

\[x^3 - 2x^2 + x = 2^3 - 2(2^2) + 2 = 8 - 8 + 2 = 2\]

This completes the first part of the proof.

For the second part, we assume that \(x^3 - 2x^2 + x = 2\) and from this assumption, we will prove that \(x = 2\). We will do this by solving this equation for \(x\). To do so, we first rewrite the equation \(x^3 - 2x^2 + x = 2\) by subtracting 2 from both sides:

\(x^3 - 2x^2 + x - 2 = 0\)

We can now factor the left side of this equation by factoring an \(x\) from the first two terms and then factoring (\(x - 2\)) from the resulting two terms. This is shown below.

\(x^2(x - 2) + x - 2 = 0\)

\((x - 2) (x^2 + 1) = 0\)

Now, in the real numbers, if a product of two factors is equal to zero, then one of the factors must be zero. So this last equation implies that

\(x - 2 = 0\) or \(x^2 + 1 = 0\)

The equation \(x^2 + 1 = 0\) has not real numbers solution. So since \(x\) is a real number, the only possibility is that \(x - 2 = 0\). From this we can conclude that \(x\) must be equal to 2.

Since we have now proven both conditional statements, we have proven that \(x = 2\) if and only if \(x^3 - 2x^2 + x = 2\)

Constructive Proofs

We all know how to solve an equation such as \(3x + 8 = 23\), where \(x\) is a real number. To do so, we first add -8 to both sides of the equation and then divide both sides of the resulting equation by 3. Doing so, we obtain the following result:

If \(x\) is a real number and \(3x + 8 = 23\), then \(x = 5\).

Notice that the process of solving the equation actually does not prove that \(x = 5\) is a solution of the equation \(3x + 8 = 23\). This process really shows that if there is a solution, then that solution must be \(x = 5\). To show that this is a solution, we use the process of substituting 5 for \(x\) in the left side of the equation as follows: If \(x = 5\),then

\(3x + 8 = 3 (5) + 8 = 15 + 8 = 23\)

This proves that \(x = 5\) is a solution of the equation \(3x + 8 = 23\). Hence, we have proven that \(x = 5\) is the only real number solution of \(3x + 8 = 23\).

We can use this same process to show that any linear equation has a real number solution. An equation of the form

\[ax + b = c\],

where \(a\), \(b\), and \(c\) are real numbers with \(a \ne 0\), is called a linear equation in one variable.

Proposition 3.12

If \(a\), \(b\), and \(c\) are real numbers with \(a \ne 0\), then the linear equation \(ax + b = c\) has exactly one real number solution, which is \(x = \dfrac{c - b}{a}\).

Therefore, the linear equation \(ax + b = c\) has exactly one real number solution and the solution is \(x = \dfrac{c - b}{a}\).

The proof given for Proposition 3.12 is called a constructive proof . This is a technique that is often used to prove a so-called existence theorem. The objective of an existence theorem is to prove that a certain mathematical object exists. That is, the goal is usually to prove a statement of the form

There exists an \(x\) such that \(P(x)\).

For a constructive proof of such a proposition, we actually name, describe, or explain how to construct some object in the universe that makes \(P(x)\) true. This is what we did in Proposition 3.12 since in the proof, we actually proved that \(\dfrac{c - b}{a}\) is a solution of the equation \(ax + b = c\). In fact, we proved that this is the only solution of this equation.

Nonconstructive Proofs

Another type of proof that is often used to prove an existence theorem is the so- called nonconstructive proof . For this type of proof, we make an argument that an object in the universal set that makes \(P(x)\) true must exist but we never construct or name the object that makes \(P(x)\) true. The advantage of a constructive proof over a nonconstructive proof is that the constructive proof will yield a procedure or algorithm for obtaining the desired object.

The proof of the Intermediate Value Theorem from calculus is an example of a nonconstructive proof. The Intermediate Value Theorem can be stated as follows:

If \(f\) is a continuous function on the closed interval [\(a\), \(b\)] and if \(q\) is any real number strictly between \(f(a)\) and \(f(b)\), then there exists a number \(c\) in the interval (\(a\), \(b\)) such that \(f(c) = q\).

The Intermediate Value Theorem can be used to prove that a solution to some equations must exist. This is shown in the next example.

Example 3.13 (Using the Intermediate Value Theorem)

Let \(x\) represent a real number. We will use the Intermediate Value Theorem to prove that the equation \(x^3 - x + 1 = 0\) has a real number solution.

To investigate solutions of thee quation \(x^3 - x + 1 = 0\), we will use the function

\(f(x) = x^3 - x + 1 = 0\)

Notice that \(f(-2)\) = -5 and that \(f(0)\) > 1. Since \(f(-2)\) < 0 and \(f(0\) > 0, the Intermediate Value Theorem tells us that there is a real number \(c\) between -2 and 0 such that \(f(c) = 0\). This means that there exists a real number \(c\) between -2 and 0 such that

\(c^3 - c + 1 = 0\),

and hence \(c\) is a real number solution of the equation \(x^3 - x + 1 = 0\). This proves that the equation \(x^3 - x + 1 = 0\) has at least one real number solution.

Notice that this proof does not tell us how to find the exact value of \(c\). It does, however, suggest a method for approximating the value of \(c\). This can be done by finding smaller and smaller intervals [\(a\), \(b\)] such that \(f(a)\) and \(f(b)\) have opposite signs.

Exercise for section 3.2

• Let \(n\) be an integer. Prove each of the following: (a) If \(n\) is even, then \(n^3\) is even. (b) If \(n^3\) is even, then \(n\) is even. (c) The integer \(n\) is even if and only if \(n^3\) is an even integer. (d) The integer \(n\) is odd if and only if \(n^3\) is an odd integer.
• In Section 3.1, we defined congruence modulo \(n\) where \(n\) is a natural number. If \(a\) and \(b\) are integers, we will use the notation \(a \not\equiv b\) (mod \(n\)) to mean that \(a\) is not congruent to \(b\) modulo \(n\). (a) Write the contrapositive of the following conditional statement: For all integers \(a\) and \(b\), if \(a \not\equiv 0\) (mod 6) and \(b \not\equiv 0\) (mod 6), then \(ab \not\equiv 0\) (mod 6). (b) Is this statement true or false? Explain.
• (a) Write the contrapositive of the following statement: For all positive real numbers \(a\) and \(b\), if \(\sqrt{ab} \ne \dfrac{a + b}{2}\), then \(a \ne b\). (b) Is this statement true or false? Prove the statement if it is true or provide a counterexample if it is false.
• Are the following statements true or false? Justify your conclusions. (a) For each \(a \in \mathbb{Z}\), if \(a \equiv 2\) (mod 5), then \(a^2 \equiv 4\) (mod 5). (b) For each \(a \in \mathbb{Z}\), if \(a^2 \equiv 4\) (mod 5), then \(a \equiv 2\) (mod 5). (c) For each \(a \in \mathbb{Z}\), \(a \equiv 2\) (mod 5) if and only if \(a^2 \equiv 4\) (mod 5).
• Is the following proposition true or false? For all integers \(a\) and \(b\), if \(ab\) is even, then \(a\) is even or \(b\) is even. Justify your conclusion by writing a proof if the proposition is true or by providing a counterexample if it is false.
• Consider the following proposition: For each integer \(a\), \(a \equiv 3\) (mod 7) if and only if \((a^2 + 5a)) \equiv 3\) (mod 7). (a) Write the proposition as the conjunction of two conditional statements. (b) Determine if the two conditional statements in Part(a) are true or false. If a conditional statement is true, write a proof, and if it is false, provide a counterexample. (c) Is the given proposition true or false? Explain.
• Consider the following proposition: For each integer \(a\), \(a \equiv 2\) (mod 8) if and only if \((a^2 + 4a) \equiv 4\) (mod 8). (a) Write the proposition as the conjunction of two conditional statements. (b) Determine if the two conditional statements in Part(a) are true or false. If a conditional statement is true, write a proof, and if it is false, provide a counterexample. (c) Is the given proposition true or false? Explain.
• For a right triangle, suppose that the hypotenuse has length \(c\) feet and the lengths of the sides are \(a\) feet and \(b\) feet. (a) What is a formula for the area of this right triangle? What is an isosceles triangle? (b) State the Pythagorean Theorem for right triangles. (c) Prove that the right triangle describe above is an isosceles triangle if and only if the area of the right triangle is \(\dfrac{1}{4}c^2\).

there exist integers \(m\) and \(n\) with \(n \ne 0\) such that \(x = \dfrac{m}{n}\).

A real number that is not a rational number is called an irrational number .It is known that if x is a positive rational number, then there exist positive integers \(m\) and \(n\) with \(n \ne 0\) such that \(x = \dfrac{m}{n}\) Is the following proposition true or false? Explain. For each positive real number \(x\), if \(x\) is irrational, then \(\sqrt x\) is irrational.

• Is the following proposition true or false? Justify your conclusion. For each integer \(n\), \(n\) is even if and only if 4 divides \(n^2\).
• Prove that for each integer \(a\), if \(a^2 - 1\) is even, then 4 divides \(a^2 - 1\).
• Prove that for all integers \(a\) and \(m\), if \(a\) and \(m\) are the lengths of the sides of a right triangle and \(m + 1\) is the length of the hypotenuse, then \(a\) is an odd integer.
• Prove the following proposition: If \(p\), \(q \in \mathbb{Q}\) with \(p < q\), then there exists an \(x \in \mathbb{Q}\) with \(p < x < q\).
• Are the following propositions true or false? Justify your conclusion. (a) There exist integers \(x\) and \(y\) such that \(4x + 6y = 2\). (b) There exist integers \(x\) and \(y\) such that \(6x + 15y = 2\). (c) There exist integers \(x\) and \(y\) such that \(6x + 15y = 9\).
• Prove that there exists a real number \(x\) such that \(x^3 - 4x^2 = 7\).
• Let \(y_1\), \(y_2\), \(y_3\), \(y_4\) be real numbers. The mean , \(\bar{y}\), of these four numbers is defined to be the sum of the four numbers divided by 4, That is, \[\bar{y} = \dfrac{y_1 + y_2 + y_3 + y_4}{4}.\] Prove that there exists a \(y_i\) with \(1 \le i \le 4\) such that \(y_i \ge \bar{y}\). Hint : One way is to let \(y_{max}\) be the largest of \(y_1\), \(y_2\), \(y_3\), \(y_4\).
• Let \(a\) and \(b\) be natural numbers such that \(a^2 = b^3\). Prove each of the propositions in Parts (6a) trough (6d). (The results of Exercise (1) and Theorem 3.10 may be helpful.) (a) If \(a\) is even, then 4 divides \(a\). (b) If 4 divides \(a\), then 4 divides \(b\). (c) If 4 divides \(b\), then 8 divides \(a\). (d) If \(a\) is even, then 8 divides \(a\). (e) Given an example of natural numbers \(a\) and \(b\) such that \(a\) is even and \(a^2 = b^3\), but \(b\) is not divisible by 8.
• Prove the following proposition: Let \(a\) and \(b\) be integers with \(a \ne 0\). If \(a\) does not divide \(b\), then the equation \(ax^3 + bx + (b + a) = 0\) does not have a solution that is a natural number. Hint: It may be necessary to factor a sum of cubes. Recall that \[u^3 + v^3 = (u + v) (u^2 - uv + v^2).\]

proposition

If \(m\) is an odd integer, then (\(m + 6\)) is an odd integer.

For \(m + 6\) to be an odd integer, there must exist an integer \(n\) such that

\[m + 6 = 2n + 1.\]

By subtracting 6 from both sides of this equation, we obtain

\[m = 2n - 6 + 1 = 2 (n - 3) = 1.\]

By the closure properties of the integers, (\(n - 3\)) is an integer, and hence, the last equation implies that \(m\) is an odd integer. This proves that if \(m\) is an odd integer, then \(m + 6\) is an odd integer.

For all integers \(m\) and \(n\), if \(mn\) is an even integer, then \(m\) is even or \(n\) is even.

For either \(m\) or \(n\) to be even, there exists an integer \(k\) such that \(m = 2k\) or \(n = 2k\). So if we multiply \(m\) and \(n\), the product will contain a factor of 2 and, hence, \(mn\) will be even.

Explorations and Activities

20. Using a Logical Equivalency. Consider the following proposition: Proposition . For all integers \(a\) and \(b\), if 3 does not divide \(a\) and 3 does not divide \(b\), then 3 does not divide the product \(a \cdot b\). (a) Notice that the hypothesis of the proposition is stated as a conjunction of two negations (“3 does not divide \(a\) and 3 does not divide \(b\)”). Also, the conclusion is stated as the negation of a sentence (“3 does not divide the product \(a \cdot b\).”). This often indicates that we should consider using a proof of the contrapositive. If we use the symbolic form \((\urcorner Q \wedge \urcorner R) \to \urcorner P\) as a model for this proposition, what is \(P\), what is \(Q\), and what is \(R\)? (b) Write a symbolic form for the contrapositive of \((\urcorner Q \wedge \urcorner R) \to \urcorner P\). (c) Write the contrapositive of the proposition as a conditional statement in English. We do not yet have all the tools needed to prove the proposition or its contrapositive. However, later in the text, we will learn that the following proposition is true. Proposition X. Let \(a\) be an integer. If 3 does not divide \(a\), then there exist integers \(x\) and \(y\) such that \(3x + ay = 1\). (d) i. Find integers \(x\) and \(y\) guaranteed by Proposition X when \(a = 5\). ii. Find integers \(x\) and \(y\) guaranteed by Proposition X when \(a = 2\). iii. Find integers \(x\) and \(y\) guaranteed by Proposition X when \(a = -2\). (e) Assume that Proposition X is true and use it to help construct a proof of the contrapositive of the given proposition. In doing so, you will most likely have to use the logical equivalency \(P \to (Q \vee R) \equiv (P \wedge \urcorner Q) \to R\).

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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.   

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A Conditional Statement Truth Table

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

Conditional Statement Examples

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

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

What does p→q represent?

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

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

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

FAQs on Conditional Statement

1. How many types of conditional statements are there?

There are basically 5 types of conditional statements.

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

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

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

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

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

Calcworkshop

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

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

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

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

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

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

Here we go!

What are Conditional Statements?

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

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

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

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

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

What is the Converse of a Statement?

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

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

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

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

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

What is the Inverse of a Statement?

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

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

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

What is a Contrapositive?

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

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

What is a Biconditional Statement?

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

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

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

Examples of Conditional Statements

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

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

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

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

Conditional Statements – Lesson & Examples (Video)

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

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2.5 Equivalent Statements

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 .

Example 2.22

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 ~ 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 .

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 . p → q ≡ ~ p ∨ q .

Your Turn 2.22

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.

Example 2.23

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 2.23

Example 2.24, identifying the converse, inverse, and contrapositive.

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 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 q : Lassie likes to bark.
• “Lassie likes to bark” is q q and “All dogs bark” is p p . So, “If Lassie likes to bark, then all dogs bark,” has the form “if q q , then p p ,” which is the form of the converse.
• “Lassie does not like to bark” is ~ q ~ q and “Some dogs do not bark” is ~ p ~ p . The statement, “If Lassie does not like to bark, then some dogs do not bark,” has the form “if ~ q ~ q , then ~ p ~ p ,” which is the form of the contrapositive.
• The contrapositive ~ q → ~ p ~ q → ~ p is logically equivalent to the conditional statement p → q . p → q .

Your Turn 2.24

Example 2.25, determining the truth value of the converse, inverse, and contrapositive.

Assume the conditional statement, p → q : 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 p : Chadwick Boseman was an actor, is true and the conclusion, q q : Chadwick Boseman did not star in the movie Black Panther , is false. The converse is q → p , 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 . ~ p → ~ q . ” The written form is: “If Chadwick Boseman was not an actor, then Chadwick Boseman starred in the movie Black Panther .” Because p p is true, and q q is false, ~ p ~ p is false, and ~ q ~ 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 ~ q → ~ p .” Because q q is false and p p is true, ~ q ~ q is true and ~ p ~ p is false. Therefore, ~ q → ~ p ~ 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 2.25

Check your understanding, section 2.5 exercises.

• Write the conditional statement p → q in words.
• Write the converse statement q → p in words.
• Write the inverse statement ~ p → ~ q in words.
• Write the contrapositive statement ~ q → ~ p in words.

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If-then statement

• Logical correct I
• Logical correct II

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

We will explain this by using an example.

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

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

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

This is noted as

$$p \to q$$

This is read - if p then q.

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

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

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

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

$$q\rightarrow p$$

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

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

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

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

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

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

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

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

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

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

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

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

This is noted:

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

If the following statements are true:

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

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

Video lesson

Write a converse, inverse and contrapositive to the conditional

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

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