 # Contrapositive and Converse

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

## What are Contrapositive Statements?

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

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

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

### What are Converse Statements?

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

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

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

Example: Consider the following conditional statement.

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

Write the contrapositive and converse of the statement.

Solution:

Given conditional statement is:

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

The converse of the above statement is:

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

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

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

Now, the contrapositive statement is:

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

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

### Contrapositive vs Converse

The differences between Contrapositive and Converse statements are tabulated below.

 Contrapositive Converse Suppose “if p, then q” is the given conditional statement “if ∼q, then ∼p” is its contrapositive statement. Note: ∼ represents the negation or inverse statement Suppose “if p, then q” is the given conditional statement “if q, then p” is its converse statement. Example: The contrapositive statement for “If a number n is even, then n2 is even” is “If n2 is not even, then n is not even. Example: The converse statement for “If a number n is even, then n2 is even” is “If a number n2 is even, then n is even. Mathematical representation: Conditional statement: p ⇒ q Contrapositive statement: ~q ⇒ ~p Mathematical representation: Conditional statement: p ⇒ q Converse statement: q ⇒ p

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

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

 p q ~p ~q ~q ⇒ ~p T T F F T T F F T F F T T F T F F T T T

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

 p q q ⇒ p T T T T F T F T F F F T

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