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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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What does $X \subset Y$ mean: "$X$ is a subset of $Y$" or "$X$ is a proper subset of $Y$"? –  Srivatsan Sep 16 '11 at 14:00
    
>If C, D are sets from a universe U, C is a subset of D, written C ⊆ D, or D ⊇ C, if every element of C is an element of D. Additionally, if D contains an element that is not in C, then C is called a proper subset of D, denoted C ⊂ D or D ⊃ C. –  a930913 Sep 16 '11 at 14:04
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The question seems ambiguous. B can be either {a, b, d} or {a, b, e}. –  Zach Langley Sep 16 '11 at 14:07
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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.) –  Srivatsan Sep 16 '11 at 14:10

3 Answers 3

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. –  a930913 Sep 16 '11 at 14:05

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 $\{a,b,d,e\}$.

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