Let X be a nonempty set and * be a binary operation on P(X), the power set of X, defined by A * B = A ∩ B for all A, B ∈ P(X).
(i) Find the identity element in P(X).
(ii) Show that X is the only invertible element in P(X).
e is the identity of * if e*a = a
From the above Venn diagram,
A*X = A∩X = A
X*A = X∩A = A
⇒ X is the identity element for binary operation *
Let B be the invertible element
⇒ A*B = X
⇒ A∩B = X
This is only possible if A = B = X
Thus X is the only invertible element in P(X)
Hence proved.