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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 of the two sets. This page may be helpful. –  Brian M. Scott Apr 14 '12 at 7:57

2 Answers 2

up vote 0 down vote accepted

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! –  Gerry Myerson Apr 14 '12 at 7:50
Thank you! am i right in saying the answer is 3 ? –  Xabi Apr 14 '12 at 7:53
@Fatz: Yes. You are right –  user9413 Apr 14 '12 at 7:57
@GerryMyerson: Are you telling me, or the OP. And perhaps the OP knows the formula, maybe he is just not able to apply. –  user9413 Apr 14 '12 at 7:59
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. –  Brian M. Scott Apr 14 '12 at 8:04

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 $y$.

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