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Thread: Proving the Cardinality of 2 finite sets

  1. #1
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    Question Proving the Cardinality of 2 finite sets

    Hi all! This is my first post, and my partner and I have struggled to try to understand the following optional/suggested problem:

    Let A and B be finite sets. Prove that |A - B| = |A u B| - |B|.
    In words: "Prove that the absolute value (or cardinality) of A minus B is equal to the absolute value of the union of A and B, minus the absolute value of B.
    On top of this we also are asked: "What does this tell us about |A|, |A - B| + |A n B| (n being the intersection of A and B).

    I have tried to set each side of the equation be equal to something, such as letting L =
    |A - B| and R = |A u B| - |B|, and showing that some arbitrary value (x) is in both L and R. For example we have: let L = |A - B| and R = |A u B| - |B|. If XeB (if x is an element of B), it is still true that XeAnB (X is an element of the intersection of A and B), because XeA.

    However, I am stuck in how this pertains to the absolute value/cardinality and how to prove this. Any help/hints would be greatly appreciated!

    (Also, sorry for the weird change of font).
    Last edited by azollner95; Feb 16th 2017 at 07:21 PM.
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  2. #2
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    Re: Proving the Cardinality of 2 finite sets

    $A \cup B =(A \cap B^c) \cup B$

    $B \cap (A \cap B^c) = \emptyset$

    $A-B = A \cap B^c$

    so

    $|A \cup B| = |A-B| + |B|$

    $|A-B| = |A \cup B| - |B|$
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