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

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} \( \boxed{123} \)

Comments

Sort by:

Top Newest

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 - 2 years, 9 months ago

Log in to reply

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 - 3 years ago

Log in to reply

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

Omkar Kulkarni - 3 years ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...