# Use of If and Then Statement

In mathematical reasoning, to infer a conclusion we frequently make use of if-then statements as:

*P: If a and b are positive integers then their product is also positive.*

This sentence can be expressed as a component of two sentences.

*a: a and b are positive integers*

*b: product of a and b is positive*

The given statement implies that the product of both the numbers is positive if it is pre-mentioned that both the numbers are positive integers.

There are certain facts we can conclude from the given implication

(i) If a is false then we cannot say anything about the truth of b

(ii) The sentence does not imply that a surely happens.

(iii) If a is true then the only b is true.

## Contrapositive statement

To make a contrapositive statement from a given conditional (if-else statement), negate both the statements and then interchange the position of the statements them. Consider the following statement:

*P: If two lines are parallel then they will have no common point of intersection.*

The component statements would be:

*a: Two lines are parallel*

*b: They will have no common point of intersection*

For making the contrapositive of this statement, we negate both the statements as following;

*~a= Two lines are parallel*

*~b=They will have a common point of intersection*

Now the contrapositive statement will be:

*S: If two lines have the common point of intersection then they are not parallel.*

The meaning delivered by the statement P is same as S.

## Converse Statement

To write the converse of a statement, both the component statements are interchanged with each other.

Consider the following statement:

*P: If any natural number n is divisible by 2 then n is even.*

The component statements would be:

*a:n is any natural number divisible by 2*

*b:n is even.*

To write the converse we interchange both the statements as follows:

*a: n is any even natural number*

*b:n is divisible by 2*

The converse statement is given as:

*S: If n is any even natural number then n is divisible by 2.*

The meaning remains the same in case we take the converse of a conditional statement.

### Example

Let us take an example and try to find the contrapositive and converse of it.

Example: *P: If two angles in a triangle are equal then the triangle is isosceles.*

Solution: The contrapositive statement is given by:

*S: A triangle is not isosceles if any two angles of a triangle are not equal*

The converse statement would be:

*S: If a triangle is isosceles then two angles of the triangle are equal*

We have seen how to use contrapositive statements and converse statements in mathematical reasoning. To learn more about mathematical reasoning, please visit us at BYJU’S.