Question

P(a intersection b) = p (a) intersection p(b)

Answer 1

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Answer 2

Answer:

To prove two sets equal, prove every element of either set is an element of the other.

Without loss of generality, let set X be an arbitrary element of P(A∩B).

Then every element of X is an element of A∩B, by the definition of a power set.

Every element of X is an element of A, by the definition of intersection.

Thus, X is an element of P(A), by the definition of a power set.

Similarly, every element of X is an element of B,

and thus X is an element of P(B).

By the definition of intersection, since X is an element of both P(A) and P(B),

X is an element of P(A) ∩ P(B).

Thus, every element of P(A∩B) is an element of P(A) ∩ P(B).

Again without loss of generality, let set Y be an arbitrary element of P(A) ∩ P(B).

By the definition of intersection, Y is an element of P(A) and Y is an element of P(B).

By the definition of a power set, every element of Y is both an element of A and an element of B.

By the definition of intersection, every element of Y is an element of A∩B.

By the definition of a power set, Y is thus an element of P(A∩B).

Thus, every element of P(A) ∩ P(B) is an element of P(A∩B).

We have shown that every element of either P(A) ∩ P(B) or P(A∩B) is an element of the other set as well. Thus, the two sets are equal.