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Let $U = \{a,b,c,...,x,y,z\}$ with $A=\{a,b,c \}$ and $C=\{a,b,d,e \}$. If $|A \cap B| = 2$ and $(A \cap B) \subset B \subset C$, determine $B$.

This question doesn't seem complete. Am I right, and if not, what is the answer?

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    a930913 Yes, the terms *subset* and *proper subset* are standard and I understand them. On the other hand, the notation $\subset$ is used to mean both "subset" and "proper subset" (depending on who uses it). That's why I commented asking you to clarify what you mean by it. (Some people, like me, use the more emphatic notations $\subseteq$ and $\subsetneq$, avoiding $\subset$ altogether.)2011-09-16

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if we are taking the subsets to be proper, then the possible answers are B={a,b,d} and B={a,b,e}, if the symbol admits subsets that are not proper, then we can also have B=C and B={a,b}. In both cases there is not a unique answer.

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Hint: $ c \not \in B$, so what is $A \cap B$? But I don't see how you can tell the difference between $d$ and $e$. If $\subset$ is proper subset you can get pretty close.

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    That's what leads me to believe that the question is incomplete.2011-09-16
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If inclusion is proper then $B=\{a,b,d\}$ or $B=\{a,b,e\}$ and if inclusion covers equality, than one could add $\{a,b\}$ and $\{a,b,d,e\}$.