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?

#Algebra

Easy Math Editor

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Comments

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, $A=B$ .

Paulo Guilherme Santos - 5 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.

Jon Haussmann - 6 years ago

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

Omkar Kulkarni - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$