Q. 133.7( 3 Votes )

Given a non-empty set X, let : P(X) × P(X) P(X) be defined as A * B = (A B) (B – A), A, B P(X). Show that the empty set φ is the identity for the operation and all the elements A of P(X) are invertible with

A–1 = A.

(Hint: (A – φ) (φ – A) = A and (A – A) (A – A) = A A = φ).

Answer :

It is given that : P(X) × P(X) P(X) be defined as

A * B = (A – B) (B – A), A, B P(X).


Now, let A ϵ P(X). Then, we get,


A * ф = (A – ф) (ф A) = A ф = A


ф * A = (ф - A) (A - ф) = ф A = A


A * ф = A = ф * A, A ϵ P(X)


Therefore, ф is the identity element for the given operation *.


Now, an element A ϵ P(X) will be invertible if there exists B ϵ P(X) such that


A * B = ф = B * A. (as ф is an identity element.)


Now, we can see that A * A = (A –A) (A A) = ф ф = ф A ϵ P(X).


Therefore, all the element A of P(X) are invertible with A-1 = A.


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