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

Previously, I raised a question whether $$ (a+F)^c=a+F^c.$$ Jonas Meyer pointed out that it is true. After which, I was able to prove the first inclusion. The details are as follows: let $y\in a+F^c$. Then there exists $z\in F^c$ such that $y=a+z$. We claim that $y\in(a+F)^c$. Suppose $y\notin (a+F)^c$. Then $y\in a+F$. Thus, there exists $x\in F$ such that $y=a+x$. This implies that $z=x$. Hence, $x\in F\cap F^c=\varnothing$. We obtain a contradiction. Thus, $y\in(a+F)^c$. Hence, $a+F^c \subseteq (a+F)^c$. I tried the other inclusion, but can't prove it. A help on this is very much appreciated.


share|cite|improve this question
up vote 1 down vote accepted

By your first step \[ (a+ F)^c = a + (-a) + (a+F)^c \subseteq a + \bigl((-a)+ a + F\bigr)^c = a + F^c. \]

share|cite|improve this answer
Ahh ok...tricky. thx very much – juniven Oct 20 '12 at 9:31

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.