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"If $A$ and $B$ are two sets, then prove that $A$ is the union of a disjoint pair of sets, one of which is contained in $B$ and one of which is disjoint from $B$."

Can somebody help me understand what this question even wants me to prove? As far as I can understand, it wants me to assume $A = C\cup D$, with $C \cap D$ being the empty set. How does $B$ even relate to this assumption, if I am not given any detail on what $B$ is? Am I supposed to prove that for ANY set $B$, $C$ would be a subset of $B$ and $D$ disjoint from $B$? I think that's a false statement, so I doubt I am reading the question correctly. Any help is greatly appreciated.

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  • $\begingroup$ $A=(A\cap B)\cup (A-B)$ $\endgroup$
    – Hanul Jeon
    Oct 13 '13 at 15:20
  • $\begingroup$ You don't have to choose $C$ and $D$ until after you learn what $B$ is .... $\endgroup$ Oct 13 '13 at 15:23
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Hint:

$$A=\left(A\cap B\right)\cup\left(A\setminus B\right)$$

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  • $\begingroup$ Thank you, this is all I needed. :) $\endgroup$
    – MM8
    Oct 13 '13 at 15:23

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