# upper and lower bound of a venn diagram

I don't quite understand upper and lower bound. To explore, I am using the following problem. Since I am not able to post an image of Venn diagram, I will try my best to explain the problem:

70% like A and 80% like B, what is the upper and lower bound of liking both A and B?

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Upper bound: everybody who likes $A$ likes $B$, so $70\%$. Lower bound: As few as possible like both. Since (at most) $A\cup B$ is everybody, that is, $100\%$, as few as possible means $50\%$: make the $20\%$ who don't like $B$ like $A$. –  André Nicolas Apr 25 '12 at 17:51
@AndréNicolas: Is it correct to assume that upper bound visually means you have 70% circle A within 80% circle B eliminating the extra 10% that only like B. And, how did you deduce 50% as few as possible? –  Vivek Apr 25 '12 at 18:44
If $A$ and $B$ have as little in common as possible, the $20\%$ space outside $B$ must be filled with people from $A$, leaving $70-20$ as overlap. For a fancier version, use $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The biggest $P(A\cap B)$ can be is $1$, when we have $P(A\cap B)=P(A)+P(B)-P(A\cup B)=0.7+0.8-1=0.5$. –  André Nicolas Apr 25 '12 at 18:48