Question:
Given a non empty set X consider the binary operation*:p(X) * p(X) - p(X) given by A*B= A intersection B Show that X is the identity element
Answer:
We have *: p(X) * p(X) -> p(X)
Given, A * B = A ∩ B
So, X * A = X ∩ A = A for all A
Hence, X is an identity element.
Again, let I is another identity.
So, I ∩ A = A for all A
and x ∈ X, I ∩ {x} = {x}
=> x ∈ I
=> X ⊂ I and I ⊂ X
=> I = X
So, X is the identity element.
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