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Thread: set question please help

  1. #1
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    set question please help

    Given sets A and B, so that |A|=m and |B|=n and |A∩B|=k, and p(S) the Power Set of S, what is the value of |p(A∪B)-(p(A) ∪ p(B)) |?

    how do i solve it.
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  2. #2
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    Re: set question please help

    What is in $p (A\cup B)$ that is not in $p (A)\cup p (B) $

    That's gonna be $p(A\Delta B)-p (A-B)-p (B-A) $ where $\Delta $ is symmetric difference.

    $|A\Delta B|=|A-B|+|B-A|=(m-k)+(n-k) $

    Can you finish from here?

    I've not checked my work, but I think the final answer is $2^{m+n-2k}-2^{m-k}-2^{n-k} $
    Last edited by SlipEternal; Apr 28th 2018 at 07:50 AM.
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  3. #3
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    Re: set question please help

    this doesn't work, you have more options then you mentioned
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    Re: set question please help

    You are right. Let's look at sets in $p (A\cup B) $ and ones in $p (A)\cup p (B) $

    So, consider the intersection of $p (A)\cap p (B) $. It is precisely $p (A\cap B) $

    So $|p (A\cup B)-(p (A)\cup p (B))|-|p (A\cup B)|-|p (A)|-|p (B)|+|p (A\cap B)|=2^{m+n}-2^m-2^n+2^k $ by Inclusion/Exclusion
    Last edited by SlipEternal; Apr 28th 2018 at 09:32 AM.
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  5. #5
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    Re: set question please help

    Quote Originally Posted by eden View Post
    Given sets A and B, so that |A|=m and |B|=n and |A∩B|=k, and p(S) the Power Set of S, what is the value of |p(A∪B)-(p(A) ∪ p(B)) |?

    how do i solve it.
    Quote Originally Posted by SlipEternal View Post
    You are right. Let's look at sets in $p (A\cup B) $ and ones in $p (A)\cup p (B) $
    So, consider the intersection of $p (A)\cap p (B) $. It is precisely $p (A\cap B) $
    So $|p (A\cup B)-(p (A)\cup p (B))|-|p (A\cup B)|-|p (A)|-|p (B)|+|p (A\cap B)|=2^{m+n}-2^m-2^n+2^k $ by Inclusion/Exclusion
    Let $A=\{a,b\}~\&~B=\{b,c\}$ thus $m=n=2~\&~k=1$

    $\mathcal{P}(A\cup B)-[\mathcal{P}(A)\cup\mathcal{P}(B)]=\{\{a,c\},\{a,b,c\}\}$

    In this example $\left|\mathcal{P}(A\cup B)-[\mathcal{P}(A)\cup\mathcal{P}(B)]\right|=2$

    Did I not understand the question?
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  6. #6
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    Re: set question please help

    Quote Originally Posted by Plato View Post
    Let $A=\{a,b\}~\&~B=\{b,c\}$ thus $m=n=2~\&~k=1$

    $\mathcal{P}(A\cup B)-[\mathcal{P}(A)\cup\mathcal{P}(B)]=\{\{a,c\},\{a,b,c\}\}$

    In this example $\left|\mathcal{P}(A\cup B)-[\mathcal{P}(A)\cup\mathcal{P}(B)]\right|=2$

    Did I not understand the question?
    Sorry, $|p(A\cup B)| = 2^{m+n-k}$. So, $|p(A\cup B) - (p(A)\cup P(B))| = 2^{m+n-k}-2^m-2^n+2^k$

    In your case, we have $2^{2+2-1}-2^2-2^2+2^1 = 2$, just as you found. I just miswrote the formula. You understood the question.
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