Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 7 down vote accepted

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

share|cite|improve this answer
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? – nilo de roock Sep 30 '15 at 8:16
@ndroock1 "I have never seen the notation Ac before." Really? – Did Sep 30 '15 at 8:32
Kindly asking for a book reference where it is used, dear Oracle of knowledge. – nilo de roock Sep 30 '15 at 8:45

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.