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I was studying for finals and I came across this question:

Assume that: $|A\cup B|=10, |A|=7$, and $|B|=6$. Determine $|A\cap B|$

How do I approach this question? I mean I know the the union must equal $10$ and $|A| +|B| =13$ but I’m lost after that.. Thank you in advance!

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    Try using a [Venn diagram](http://en.wikipedia.org/wiki/Venn_diagram) of the two sets. [This page](http://www.purplemath.com/modules/venndiag4.htm) may be helpful. – 2012-04-14

2 Answers 2

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Use the following formula:

  • $|A \cup B| = |A| + |B| - |A \cap B| \Rightarrow |A\cap B| = |A|+|B| - |A \cup B|$
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    @GerryMyerson,Brian : It's ok. $D$ifferent ways of giving an answer. I too agree with the fact that the OP might benefit with Gerry's answer. – 2012-04-14
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Write $x$ for the part of $A$ that's not in $B$, $y$ for the part of $A$ that is in $B$, and $z$ for the part of $B$ that's not in $A$. Then you are given $x+y+z$, $x+y$, and $y+z$, and you are asked for $y$.