Question

Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B &mnForE; A, B in P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation*.

Answer 1

It is given that .:`P(X) x P(X) -> P(X)` is defined as `A * B = A "∩" B ∀ A, B in  P(X) `

We know that  A ∩ X =  A = X ∩ A ∀ A ∈ P(X)

`=> A * X = A = X * A ∀ A ∈ P(X)` 

Thus, X is the identity element for the given binary operation *.

Now, an elementis A ∈ P(X)invertible if there exists B ∈ P(X) such that

A * B = X = B * A (As X is the identity element)

i.e

A ∩ B = X = B ∩ A

This case is possible only when A = X = B.

Thus, X is the only invertible element in P(X) with respect to the given operation*.

Hence, the given result is proved.