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

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

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 ∀ a, b ∈ A.

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

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

According to the definition of the operator, we have

⇒ a * b = ab 

We know that a^b ∈ N ∀ a, b ∈ N.

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

Hence, statement 1 is true.

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

Let a = 2, b = - 3 ∈ Z and operation * is defined above.

According to the definition of the operator, we have

⇒ a * b = ab = 2^{- 3} = 1/8

We know that 1/8 is not an integer i.e a * b ∉ Z.

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

Hence, statement 2 is false.

So, option A is the correct answer.