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

Which of the following statements is/are true: Let * be a binary operation on a non-empty set A such that a * b = a/b If A = Z then * is a binary operation on Z. If A = N then * is a binary operation on N. Correct Answer Neither 1 nor 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 = a/b

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

Let a = 1, b = 2 ∈ Z and operation * is defined above.

According to the definition of the operator, we have

⇒ a * b = 1/2

We know that, 1/2 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 = N then * is a binary operation on N.

Let a = 1, b = 2 ∈ N and operation * is defined above.

According to the definition of the operator, we have

⇒ a * b = 1/2

We know that, 1/2 is not a natural number. i.e a * b ∉ N

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

Hence, statement 2 is also false.

So, option D is the correct answer.

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