Question

Prove that A is in the subset B iff P(A) is in the subset P(B)

Answer 1

\[A \subseteq B\] iff \[P(A) \subseteq P(B)\]

Answer 2

\(P(A)\) is the power set of \(A\) correct?

Answer 3

Correct.

Answer 4

Since it's an iff statement, we need to prove implication in both directions. Let's start with \[A \subseteq B \Rightarrow P(A)\subseteq P(B)\]If we let some element \(a\in A\) that means that \(a\in B\) as well. The same is true for any subset of \(A\). So\[A' \subseteq A\Rightarrow A' \subseteq B\]Since the power set is defined as the set of all subsets, this means that \[A'\subseteq A \Rightarrow A' \subseteq P(A)\]and that\[A' \subseteq A\Rightarrow A' \subseteq P(B)\]This implies that \[P(A)\subseteq P(B)\]
Now we still need to do the other way.

Answer 5

Suppose \(P(A)\subseteq P(B)\). Then we know that \(a\in P(A) \Rightarrow a\in P(B)\). However, since \(a\in P(A)\), this also implies that \(a\) is a subset of \(A\). Also, since \(a\in P(B)\) this implies that \(a\) is a subset of \(B\).

Thus, it must be that \[A \subseteq B\]

Answer 6

You are awesome kingGeorge.

Answer 7

Why thank you. :)