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), 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 * .

Solution:

Leave T alone a non-void set and P(T) be its force set. Let any two subsets An and B of T. A ∪ B ⊂ T

Along these lines, A ∪ B ∈ P(T)

Thusly, ∪ is a twofold procedure on P(T).

Additionally, assuming A, B ∈ P(T) and A – B ∈ P(T), the convergence of sets and contrast of sets are likewise double procedure on P(T) and A ∩ T = A = T ∩ A for each subset An of sets

A ∩ T = A = T ∩ A for every one of the A ∈ P(T)

T is the character component for convergence on P(T).