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 the operation

∗:P( X )×P( X )→P( X ) is defined as A∗B=A∩B, where A,B∈P( X ).

Since,

A∩X=A X∩A=A

Therefore, by the given relation,

A∗X=A X∗A=A

So, X is the identity element for the given binary operation.

An element A∈P( X ) is invertible if there exist B∈P( X ) which follows the given condition,

A∗B=X=B∗A A∩B=X=B∩A

The above condition is only possible when A=X and B=X. So, X is the only invertible element in P( X ) with respect to the given operation ∗.

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