Question

Given a non-empty set X, consider the binary operation * : P (X)×P (X)→P (X) given by A * B= A∩B ∀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) × P (X) \(\to\) P (X) is defined as A * B = A ∩ B ∀ A, B ∈ P(X)
We know that A ∩ X = A = X ∩ A  ∀ A ∈ P (X)
\(\Rightarrow\) A * X = A = X * A  ∀ A ∈ P (X)
Thus, X is the identity element for the given binary operation *
Now, an element A ∈ P (X) is invertible if there exists B ∈ P(X) 
such that A * B = X = B * A  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.