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

∗: P(X) × P(X) → P(X) given by
A ∗ B = A ∩ B ∀ A, B ϵ P(X).
As we know that,
 A * X = A = X * A $\forall$ A ϵ P(X).
Thus, X is the identity element for the given binary operation *.
Now, an element A ϵ P(X) is invertible if there exists B ϵ P(X) such that
A * B = X = B * A (As X is the identity element)
A ∩ B = X = B ∩ A
$\textbf{\underline{This can be possible only when A = X = B.}}$

$\textbf{Therefore, X is the only invertible}\\\textbf{ element in P(X) w.r.t. given operation *.}$