question_answer 65)

Given a non empty set X, consider the binary operation * : P(X) × P(X)  given by A * B =        where 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:

. Given * P (X) × P(X)  defined as             Let, if possible, I be an identity elements in P(X).                                      Also                                 and I and       Thus, X is the identity element. 2nd Part :  Let A be an invertible element of P(X).        there exist an element  such that                                                  A = X and B = X.        is the only invertible element in P(X) and       X?1 = X.