Converse of a Statement

Generally, a compound statement is a combination of two or more simple statements. The simple statements can be converted into compound statements using multiple ways. In Mathematics, we have seen many theorem statements which are connected using the words “if” and “then”. Now, let us discuss how to find the converse of a compound statement that uses the words “if” and “then” with the help of examples.

Converse of a Statement Definition

Let P and Q be the two simple statements, and P ⇒ Q be the compound statement.

Therefore, the converse of a statement P ⇒ Q is Q ⇒ P.

It should be observed that P ⇒ Q and Q ⇒ P are converse of each other.

In Geometry, we have come across the situations where P ⇒ Q is true, and we have to decide if the converse, i.e., Q ⇒ P, is also true.

Now, let us understand how to find the converse of a statement with the help of an example.

Given statement: If a triangle ABC is an equilateral triangle, then all its interior angles are equal.

To find the converse of a given statement, first we have to identify the statements P and Q.

The given statement is in the form P ⇒ Q. Now, we have to find Q ⇒ P.

Here, P = Triangle ABC is an equilateral triangle.

Q = Triangle ABC interior angles are equal.

Hence, the converse of a statement is “If all the interior angles of triangle ABC are equal, then it is an equilateral triangle”, which is in the form Q ⇒ P.

Converse of a Statement Examples

Go through the following examples to find the converse of a statement.

Example 1:

Find the converse of a statement: If Ashok is riding a bicycle, then 17th August falls on a Sunday.

Solution:

Given compound statement: If Ashok is riding a bicycle, then 17th August falls on a Sunday (The statement is in the form P ⇒ Q)

Here, P = Ashok is riding a bicycle

Q = 17th August falls on a Sunday.

Hence, the converse of a statement, Q ⇒ P is “If 17 August falls on a Sunday, then Ashok is riding a bicycle”.

Example 2:

Determine the converse of a statement: If the decimal expansion of a real number is terminating, then the number is rational.

Solution:

Given compound statement ( P ⇒ Q): If the decimal expansion of a real number is terminating, then the number is rational.

Here,

P = The decimal expansion of a real number is terminating.

Q = The real number is rational.

Hence, the converse of a statement (Q ⇒ P) is “If a real number is rational, then its decimal expansion is terminating”.

Example 3:

State the converse of a statement: “If n is an even integer, then 2n+1 is an odd integer”.

Solution:

Given: P ⇒ Q is “If n is an even integer, then 2n+1 is an odd integer”.

Here,

P = n is an even integer.

Q = 2n+1 is an odd integer.

Therefore, the converse of a statement (i.e) Q ⇒ P is “If 2n+1 is an odd integer, then n is an even integer”.

Practice Problems on the Converse of a Statement

Find the converse of the following statements:

  1. If the water tank is black, then it contains potable water.
  2. If two triangles are congruent, then their corresponding angles are equal.
  3. If an animal is a dog, then it has a tail.

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