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Let $X$ and $Y$ be two arbitrary subsets of $\mathbb{R}$. Show that $\overline{X\cap Y} \subseteq \overline{X} \cap \overline{Y}$

Proof

since $X\cap Y \subseteq X$ and $X\cap Y \subseteq Y$

$\implies \overline{X\cap Y} \subseteq \overline{X}$ and $\overline{X\cap Y} \subseteq \overline{Y}$

In other words, $\overline{X\cap Y}$ is present in both $\overline{X}$ and $\overline{Y}$. $$\implies \overline{X\cap Y} \subseteq \overline{X} \cap \overline{Y}$$ Is my proof correct?

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    $\begingroup$ It is a quite good proof. $\endgroup$
    – Jihad
    Dec 31, 2014 at 12:58
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    $\begingroup$ It seems correct yes $\endgroup$
    – user169373
    Dec 31, 2014 at 12:58
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    $\begingroup$ Am I missing something? The overline indicates complement set within the given universal set, correct? So then wouldn't we have $\overline X\subseteq \overline {X\cap Y}$ instead of the other way around? $\endgroup$
    – abiessu
    Dec 31, 2014 at 13:04
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    $\begingroup$ Excellent. But I would leave out the "other words". Saying that $A$ is a subset of $B$ is okay and enough. Saying that $A$ is present in $B$ raises questions like: what is meant by that? $\endgroup$
    – drhab
    Dec 31, 2014 at 13:04
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    $\begingroup$ @abiessu Here overline indicates closure. $\endgroup$
    – drhab
    Dec 31, 2014 at 13:06

1 Answer 1

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Yes.

There's really not much more to say about it. The steps are trivial and clear. One might even regard the whole statement as obvious.

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