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

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? – 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
In a purely logical sense, it means the union of $B$ and the complement of $A$. – Thomas Andrews Dec 27 '11 at 15:48
@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

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
I have never seen the notation $A^c$ before. Where is it used? – ndroock1 Sep 30 at 8:16
@ndroock1 "I have never seen the notation Ac before." Really? – Did Sep 30 at 8:32
Kindly asking for a book reference where it is used, dear Oracle of knowledge. – ndroock1 Sep 30 at 8:45