Book of Proof 2nd Edition Chapter 9 Page 153 Prove or Disp

(Book of Proof 2nd Edition // Chapter 9 // Page 153)

Prove or Disprove the following:

If A and B are sets, then (A) intersection (B) = (A intersection B).

Solution

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 AB.

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

Let set X be an arbitrary element of P(AB).
Then every element of X is an element of AB, 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(AB) is an element of P(A) P(B).

Hence, every element of either P(A) P(B) or P(AB) is an element of the other set as well. Thus, the two sets are equal.