Contrapositive Statements

A contrapositive is a form of a conditional statement. It is an outcome statement after exchanging the hypothesis and conclusion of an inverse statement, as the inverse statement is a must in calculating the contrapositive statement.

Therefore, we first need to find the inverse statement of any conditional statement present. Now, all we have to do is swap conclusions and hypotheses to get this statement. For example, an inverse statement like, “If it is not a Monday, then I will not wake up late.” The contrapositive statement shall be, “If I am not waking up late, then it is not a Monday.”

Conditional Statement

To understand the contrapositive statement, we need to know about the conditional statement and its converse and inverse too. 

A conditional statement carries a hypothesis with a conclusion. It shall be like “if p, then q,” where p is the hypothesis for any situation and q is the conclusion of that hypothesis.

A conditional statement shall be like, “if p happens, the q will happen”. Here, the ‘p’ represents any hypothesis, whereas the ‘q’ represents the outcome or conclusion of that hypothesis.

For example, “if she will decline the work, then she will have to leave the institute”.

Here, the first part, “If she will decline the work,” is the hypothesis, whereas the second part, i.e., “then she will have to leave the institute,” is the conclusion. Another example of a conditional statement is “If today is Thursday, then yesterday was Wednesday”.

There can be different creations by the conditional statement; converse, inverse, and contrapositive statements. 

• The Converse of Statement: Assume any statement as p→q. Where p is considered as the hypothesis and q as the conclusion of the statement. In the converse statement, we interchange these two, i.e., the hypothesis and conclusion from the original statement. This shall now be as, q→ p. 

Example: A conditional statement- If today is Thursday, then tomorrow will be Friday.

Hypothesis: If today is Thursday

Conclusion: then tomorrow will be Friday

The converse statement- If tomorrow will be Friday, then today is Thursday.

• The Inverse of Statement: In an inverse statement, both the hypothesis and the conclusion of the original conditional statement are negative. The representation of an inverse statement shall be as ~p(not p)→ ~q (not q).

Example: A conditional statement- If today is Thursday, then tomorrow will be Friday.

Hypothesis: If today is Thursday

Conclusion: then tomorrow will be Friday

Inverse Statement: If today is not Thursday, then tomorrow will not be Friday.

• The Contrapositive Statements: For creating the contrapositive statements, we need to interchange both the conclusion and hypothesis of the inverse statement. Thus, as mentioned above, the determination of the inverse statement is a must for the contrapositive statement. 

Inverse Statement: If today is not Thursday, then tomorrow will not be Friday.

Hypothesis: If today is not Thursday

Conclusion: then tomorrow will not be Friday.

The Contrapositive Statement: If tomorrow will not be Friday, then today is not Thursday.

Examples of the Contraceptive Statements

1- If a triangle is equilateral, then they have the same sides?

A1– For determining the contrapositive statement, we will need the inverse statement. 

Inverse Statement: If the triangle is not equilateral, then it does not have the same sides.

Contrapositive statement: If it does not have the same sides, then it is not equilateral.

 2- What is the contrapositive of the inverse of p ⇒ ~q?

A2– As we know, the inverse of p ⇒ ∼q is,

 ∼ p ⇒ q

Therefore, the contrapositive of ∼ p ⇒ q shall be ∼ q ⇒ p. 

 3- Write the contrapositive statement for the following: “If you study well, then you will be a topper”.

A3– Inverse statement: If you do not study well, then you will not be a topper.

Contrapositive statement: If you are not a topper, then you do not study well. 

4- If a quadrilateral is a square, then it has 4 equal angles.

A4-Inverse Statement: If a quadrilateral is not square, then it does not have 4 equal angles.

Contrapositive Statement: If a quadrilateral does not have 4 equal angles, then it is not a square.

Conclusion 

A contrapositive is a form of a conditional statement. It is an outcome statement after exchanging the hypothesis and conclusion of an inverse statement, as the inverse statement is a must in calculating the contrapositive statement. A conditional statement carries a hypothesis with a conclusion. A conditional statement shall be like, “if p happens, the q will happen”.

Here, the ‘p’ represents any hypothesis, whereas the ‘q’ represents the outcome or conclusion of that hypothesis. The contrapositive statement is the outcome of a conditional statement itself. Another type of conditional statement instead of the inverse and contraceptive is the converse statement.