# Set theory proofs

Could someone help me out with these set theory proofs? $$A$$, $$B$$ and $$C$$ are sets, and the questions are independent of each other.

Q1. Show that if $$A\subset B$$, then $$C-B\subset C-A$$.

Q2. If $$P(A)=P(B)$$, show that $$A=B$$.

Note: $$P(A)$$ denotes the power set of $$A$$.

Also, could you provide me an faster solution to this question? The question is 'Show that $$A=(A\cup B)\cap(A-B)$$'. My method is to assume that $$a\in A$$, and prove that $$a\in (A\cup B)\cap(A-B)$$, and hence say that $$A\subseteq(A\cup B)\cap(A-B)$$, and prove vice versa. I find this method extremely lengthy. Is there a faster method?

Note by Omkar Kulkarni
3 years ago

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Q3: Let's suppose that $$\mathscr{P}(A)=\mathscr{P}(B)$$.

Let $$x \in A$$. Then $$\exists C \in \mathscr{P}(A) : x \in C$$. Let $$C \in \mathscr{P}(A)$$ be in that conditions. By hypothesis, we have that $$C \in \mathscr{P}(B)$$, this means that $$x \in C \subseteq B$$, so $$x \in B$$.

Now, $$y \in B$$. Then $$\exists D \in \mathscr{P}(B) : x \in D$$. Let $$D \in \mathscr{P}(B)$$ be in that conditions. By hypothesis, we have that $$D \in \mathscr{P}(A)$$, this means that $$y \in D \subseteq A$$, so $$y \in A$$.

In conclusion, $$A=B$$.

- 2 years, 9 months ago

Q1: Show that if $$x \in C \setminus B$$, then $$x \in C \setminus A$$.

Q2: This basically says that if you are given $$P(A)$$, then you can uniquely determine the set $$A$$. For example, if $P(A) = \{\emptyset, \{1\}, \{4\}, \{5\}, \{1,4\}, \{1,5\}, \{4,5\}, \{1,4,5\}\},$ then what is $$A$$? You just need to generalize this idea.

As for your last question, I don't think there's a faster way. That's the traditional approach for showing that two sets are equal. Sometimes, you may be able to use some identity that you have proven before, but I doubt that's the cas here.

- 3 years ago

Oh okay. Makes sense. Thank you! I'll come back to you with the second one if I don't get it.

- 3 years ago