Converse Statement

Converse Statement

A statement that is of the form "If p then q" is a conditional statement. Here 'p' refers to 'hypotheses' 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. The converse statement is " If Cliff drinks water then she is thirsty". 

What is converse statement?

A converse statement is the opposite of a conditional statement. It is to be noted that not always the converse of a conditional statement is true. 

For example, in geometry, "If a closed shape has four sides then it is a square" is a conditional statement, The truthfulness of a converse statement depends on the truth of hypotheses of the conditional statement.

In this mini-lesson, we will learn about the converse statement, how inverse and contrapositive are obtained from a conditional statement, converse statement definition, converse statement geometry, and converse statement symbol.

Lesson Plan

What Is Converse Statement?

Definition

A statement obtained by reversing the hypothesis and conclusion of a conditional statement is called a converse statement.

Explanation

Let us understand the terms "hypothesis" and "conclusion."

A statement which is of the form of "if p then q" is a conditional statement, where 'p' is called hypothesis and 'q' is called the conclusion.

A converse statement is gotten by exchanging the positions of 'p' and 'q' in the given condition.

if \(\begin{align}  p \rightarrow  q,\end{align}\) then, \(\begin{align}  q \rightarrow  p\end{align}\)

For example,

"If Cliff is thirsty, then she drinks water" is a condition.

The converse statement is "If Cliff drinks water, then she is thirsty."

Converse statement

The hypothesis 'p' and conclusion 'q' interchange their places in a converse statement.


What Is Inverse Statement?

Definition

A statement obtained by negating the hypothesis and conclusion of a conditional statement.

Explanation

An inverse statement changes the "if p then q" statement to the form of  "if not p then not q."

if \(\begin{align} p \rightarrow q,\end{align}\) then, \(\begin{align}  \sim{p} \rightarrow  \sim{q}\end{align}\)

For example,

"If John has time, then he works out in the gym."

The inverse statement is "If John does not have time, then he does not work out in the gym." 


What Is Contrapositive Statement?

Definition

A statement obtained by exchanging the hypothesis and conclusion of an inverse statement. 

Contrapositive statement

Explanation

A contrapositive statement changes "if not p then not q" to "if not q to then, not p."

  • The converse of the conditional statement is “If Q then P.”
  • The contrapositive of the conditional statement is “If not Q then not P.”
  • The inverse of the conditional statement is “If not P then not Q.”
if \(\begin{align} p \rightarrow q,\end{align}\) then, \(\begin{align}  \sim{q} \rightarrow  \sim{p}\end{align}\)

For example,

If it is a holiday, then I will wake up late. - Conditional statement

If it is not a holiday, then I will not wake up late. - Inverse statement

If I am not waking up late, then it is not a holiday. - Contrapositive statement


What Is a Conditional Statement? 

Definition

A conditional statement is a statement in the form of "if p then q," where 'p' and 'q' are called a hypothesis and conclusion.

conditional statement

A conditional statement defines that if the hypothesis is true then the conclusion is true.

For example,

"If we have to to travel for a long distance, then we have to take a taxi" is a conditional statement.


Converse of a Conditional Statement

To get the converse of a conditional statement, interchange the places of hypothesis and conclusion.

if \(\begin{align} p \rightarrow q,\end{align}\) then, \(\begin{align}  q \rightarrow  p\end{align}\)

For example,

If you eat a lot of vegetables, then you will be healthy. - Conditional statement

If you are healthy, then you eat a lot of vegetables. - Converse of Conditional statement


Inverse of Conditional Statement

To get the inverse of a conditional statement, we negate both the hypothesis and conclusion.

if \(\begin{align} p \rightarrow q,\end{align}\) then, \(\begin{align}  \sim{p} \rightarrow  \sim{q}\end{align}\)

For example,

If you read books, then you will gain knowledge. - Conditional statement

If you do not read books, then you will not gain knowledge. -Inverse of conditional statement


Contrapositive of Conditional Statement

To get the contrapositive of a conditional statement, we negate the hypothesis and conclusion and exchange their position.

if \(\begin{align} p \rightarrow q,\end{align}\) then, \(\begin{align}  \sim{q} \rightarrow  \sim{p}\end{align}\)

For example,

Emily's dad watches a movie if he has time. - Conditional statement

If Emily's dad does not have time, then he does not watch a movie. - Contrapositive of a conditional statement

 
important notes to remember
Important Notes
  1. In a conditional statement "if p then q," 'p' is called the hypothesis and 'q' is called the conclusion.
  2. There can be three related logical statements for a conditional statement.

Solved Examples

Example 1

 

 

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

If you win the race then you will get a prize.

Solution

The conditional statement given is "If you win the race then you will get a prize." 

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

Here 'p' is the hypothesis and 'q' is the conclusion.

Converse statement is "If you get a prize then you won the race." (If q then p)

Inverse statement is "If you do not win the race then you will not get a prize." (If not p, then not q)

Contrapositive statement is "If you did not get a prize then you did not win the race ." (if not q then  not p)

Example 2

 

 

From the given inverse statement, write down its conditional and contrapositive statements.

If there is no accomodation in the hotel, then we are not going on a vacation.

Solution

The inverse statement given is "If there is no accomodation in the hotel, then we are not going on a vacation."

It is of the form "If not p then not q"

Here 'p' is the hypothesis and 'q' is the conclusion.

Conditional statment is "If there is accomodation in the hotel, then we will go on a vacation." (If p then q)

Contrapositive statement is "If we are not going on a vacation, then there is no accomodation in the hotel." (If not q then not p)

 

Example 3

 

 

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

If you study well then you will pass the exam.

Solution

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)

 
Thinking out of the box
Think Tank
 
 
 

    converse, inverse, and contrapositive

  1. If 2a + 3 <  10, then a = 3. Write the converse, inverse, and contrapositive statements and verify their truthfulness.

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 converse statement. Hope you enjoyed learning! Now you can easily find the converse, inverse, and contrapositive of any conditional statement you are given!

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Frequently Asked Questions (FAQs)

1. What is the negation of a statement?

A statement that conveys the opposite meaning of a statement is called its negation.

2. How do you write a converse statement?

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

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