Which of the following statements is/are true:
Let S be a non-empty set and P(S) be its power set.
1. Union on P(S) is a binary operation.
2. Intersection on P(S) is not a binary operation.

Question

Question

Answer 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: S is a non-empty set and P(S) be its power set

Statement 1: Union on P(S) is a binary operation.

Let A, B ∈ P(S) i.e A ⊆ S and B ⊆ S

⇒ A ∪ B ⊆ S ---------(∵ A ⊆ S and B ⊆ S)

⇒ A ∪ B ∈ P(S)

So, P(S) is closed with respect to operation union.

Hence, Statement 1 is true.

Statement 2: Intersection on P(S) is not a binary operation.

Let A, B ∈ P(S) i.e A ⊆ S and B ⊆ S

⇒ A ∩ B ⊆ S----------(∵ A ⊆ S and B ⊆ S)

⇒ A ∩ B ∈ P(S)

So, P(S) is closed with respect to operation intersection.

Hence, statement 2 is false.

So, option A is the correct answer.