# how can I show that $\overline A + \overline B = \overline{A+B}$ [duplicate]

Given $A$ and $B$ two non empty set in $\mathbb R$ with $A$ bounded how can I show that $$\overline A + \overline B = \overline{A+B}$$

I have no idea how to approach this question.

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## marked as duplicate by Asaf Karagila, amWhy, Marvis, Brian M. Scott, martiniNov 23 '12 at 6:28

This question was marked as an exact duplicate of an existing question.

What does $\bar{A}$ denote? Does it denote closure (or) complement? And I assume $C+D$ stands for $\{x+y: x \in C \text{ and }y \in D\}$. – user17762 Nov 23 '12 at 0:01
Not every question about sets has to do with set theory. – Asaf Karagila Nov 23 '12 at 0:03
@Marvis $\overline A$ is the closure – Prince Nov 23 '12 at 0:10
Equality of sets can be viewed as two inclusions. Can you do one of the inclusions: $\overline A + \overline B \subseteq \overline{A+B}$ or $\overline A + \overline B \supseteq \overline{A+B}$ ?? – GEdgar Nov 23 '12 at 0:14
@GEdgar no, i can't – Prince Nov 23 '12 at 0:16

The above link provides an example of sets A and B which are closed such that A+B is not closed. Using the above, we'd have that $\bar{A} = A$, and $\bar{B} = B$, but since $A +B$ is not closed, the proposition fails.
@amWhy: commenting is only possible when reputation is $\geq50$. (and thanks for the dup. link in the closure!) – Asaf Karagila Nov 23 '12 at 0:31