Q. 9 Given a non-empty set X, consider the binary operation * : P(X) \times P(X) \rightarrow P(X)
given by A * B = A \cap B\;\; \forall A ,B\in P(X), 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 ∗.

Answers (1)

Given   * : P(X) \times P(X) \rightarrow P(X)   is defined as  A * B = A \cap B\;\; \forall A ,B\in P(X).

As we know that A \cap X=A=X\cap A\forall A \in P(X)

 \Rightarrow            A*X =A=X*A \forall A in P(X)

  Hence, X is the identity element of binary operation *.

Now, an element A \in P(X) is invertible if there exists a B \in P(X) ,

such that     A*B=X=B*A         (X is identity element)

  i.e.             A\cap B=X=B\cap A

This is possible only if  A=X=B.

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

                                                                                                   

 

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