Which of the following statements is/are true: Let * be a binary operation on a non-empty set A such that a * b = ab/2 1. If A = Z then * is a binary operation on Z. 2. If A = R then * is a binary operation on R.

Which of the following statements is/are true: Let * be a binary operation on a non-empty set A such that a * b = ab/2 1. If A = Z then * is a binary operation on Z. 2. If A = R then * is a binary operation on R. Correct Answer Only 2

Concept:

An operation * on a non-empty set S, is said to be a binary operation if it satisfies the closure property.

Closure Property:

Let S be a non-empty set and a, b ∈ S, if a * b ∈ S for all a, b ∈ S then S is said to be closed with respect to operation *.

Calculation:

Given: * is a binary operation on a non-empty set A such that a * b = ab/2

Statement 1: If A = Z then * is a binary operation on Z.

Let a = 1, b = 3 ∈ Z and operation * is defined above.

According to the definition of the operator, we have

⇒ a * b = (1 × 3) / 2 = 3/ 2

We know that, 3/2 is not an integer i.e a * b ∉ Z.

So, Z is not closed with respect to the given operation *

Hence, statement 1 is false.

Statement 2: If A = R then * is a binary operation on R.

Let a, b ∈ R and operation * is defined above.

According to the definition of the operator, we have

⇒ a * b = ab/2

We know that ab/2 is a real number i.e a * b ∈ R.

So, R is closed with respect to the given operation *

Hence, statement 2 is true.

So, option B is the correct answer.