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How to interpret the set $\{x\mid x \in A \implies x \in B \}$?

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?2011-12-27
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    Surely $\{x|x\in A \subset B\}$. I don't see the big problem.2011-12-27
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    In a purely logical sense, it means the union of $B$ and the complement of $A$.2011-12-27
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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.2011-12-27

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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.2011-12-27
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    This makes a lot more sense, thank you.2011-12-27
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    I have never seen the notation $A^c$ before. Where is it used?2015-09-30
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    @ndroock1 "I have never seen the notation Ac before." [Really?](https://en.wikipedia.org/wiki/Complement_(set_theory))2015-09-30
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    Kindly asking for a book reference where it is used, dear Oracle of knowledge.2015-09-30
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    @ndroock1 Did you notice the link in my previous comment?2015-09-30