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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

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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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    But before you use it, try to understand why it is true!2012-04-14
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    Thank you! am i right in saying the answer is 3 ?2012-04-14
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    @Fatz: Yes. You are right2012-04-14
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    @GerryMyerson: Are you telling me, or the OP. And perhaps the OP knows the formula, maybe he is just not able to apply.2012-04-14
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    I can’t speak for @Gerry, but I seconded his comment partly for the benefit of the OP and partly to suggest that this answer isn’t as useful as one that that deals more directly with the ideas involved.2012-04-14
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    I'm sure, Chandrasekhar, that you understand why the formula is true.2012-04-14
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    Any two non empty sets may or may not have common elements. In your problem, A and B have common elements.2012-04-14
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    @GerryMyerson,Brian : It's ok. Different 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 $y$.