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I've seen it in exercises from a few texts, but it isn't obvious to me. Thanks.

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Which texts for instance? –  lhf Dec 27 '11 at 15:41
    
Surely $\{x|x\in A \subset B\}$. I don't see the big problem. –  simplicity Dec 27 '11 at 15:46
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In a purely logical sense, it means the union of $B$ and the complement of $A$. –  Thomas Andrews Dec 27 '11 at 15:48
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@simplicity I doubt it. That notation makes it seem like $A\subset B$ which implies the set is just $A$ whereas in the original notation it just has to make sense to take the intersection. –  Matt Dec 27 '11 at 15:49

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up vote 6 down vote accepted

This (unconventionally) defines the set $$B\cup(A^c).$$ Hint: the assertion $P\implies Q$ is equivalent to $Q\lor(\lnot P)$.

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Of course, if $A$ and $B$ are sets, and we aren't implicitly working inside some ambient set, then $B \cup (A^c)$ will not actually be a set, but rather a proper class. –  Chris Eagle Dec 27 '11 at 15:56
    
This makes a lot more sense, thank you. –  bill billerson Dec 27 '11 at 16:12

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