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Suppose that A is a subset of B. How can we show that B-complement is a subset of A-complement?

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tried specific exmaples, but couldn't find a general solution. – Baha Jan 9 '11 at 3:25
A belongs to? Maybe you mean subset? – Aryabhata Jan 9 '11 at 3:37
yes, that's what I mean – Baha Jan 9 '11 at 14:26
up vote 4 down vote accepted

Suppose $b\in B^c$. Is it possible that $b\in A$? Suppose it is, and derive a contradiction, and you should have your desired containment.

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no, because A belongs to B, but b belongs to B-complement. How did you write with math notations? – Baha Jan 9 '11 at 3:29
This site uses TeX markup, my specific code is $b\in B^c$ and $b\in A$. To get them to look the way they do, surround them with $ signs. Here is a good list of basic symbols. – yunone Jan 9 '11 at 3:33

Instead of using a proof by contradiction, $b\in B^c \implies b \notin A$ since $A \subset B$ hence $b \in A^c$ which implies the result.

The key here being that for all $x$, either $x\in X$ or $x\in X^c$.

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