Question

Given a non-empty set X, consider the binary operaion * : P(X) x P(X) → P(X), given by A * B = A ∩ B, for all 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 *.

Answer 1

We have * : P(X) x P(X) → P(X)

given by A* B = A ∩B

∴ X * A = X ∩ A = A * X = A for all A.

∴ X is an identity element.

Let I is an other identity

⇒ I ∩ A = A ∩ I = A for all A

and x ∈ X, I ∩ {x} = {x}

x ∈ I ⇒ X ⊂ I and I ⊂ X

⇒ I = X.