proving 2 sets?

Question

how do you show that if A and B are sets such that power set of A is an element of the power set B, then A is an element of B

Answer 1

Let P(A) denote the power set of A. Then you are assuming that P(A) is an element of P(B). This means that P(A) is a subset of B (that's what it means to be an element of P(B)). Hence, every element of P(A) is an element of B (that's what it means to be a subset). But since A is an element of P(A) (since A is a subset of A), we get that A is an element of B.

Answer 2

The power set of B contains all the subsets of B.

The power set of A is an element of the power set of B

---> The power set of A is a subset of B.

---> All elements of the power set of A are elements of B

A is a subset of A, and therefore A is an element of the power set of A.

Therefore A is an element of B.

Answer 3

A is an element of B, or A is a subset of B? You cannot show that it is an element of B, but you can show that it is a subset of B.

If the power set of A is an element of the power set of B, then for any subset of A, that set is contained in B. Since A is a subset of A, the theorem is proved.

This is one of those questions for which writing out the definitions (which I did not, but assume that you can) completes most of the problem for you.

Answer 4

The power set of B, which I'll abbreviate as P(B), is the collection of all subsets of B. An element of P(B) is a subset of B. If P(A) is an element of P(B), then P(A) is a subset of B. That means any element of P(A) is also an element of B.

P(A) is the collection of all subsets of A. A itself is one of these subsets, so A is an element of P(A). Since every element of P(A) is an element of B, we conclude that A is an element of B.

Answer 5

If B contains all the powers of A, then B must contain all A^1