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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?

MAS
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    It is a quite good proof. – Jihad Dec 31 '14 at 12:58
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    It seems correct yes –  Dec 31 '14 at 12:58
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    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? – abiessu Dec 31 '14 at 13:04
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    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? – drhab Dec 31 '14 at 13:04
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    @abiessu Here overline indicates closure. – drhab Dec 31 '14 at 13:06
  • @drhab: ah, thank you. I missed the implication hinted at by the topology tag. – abiessu Dec 31 '14 at 13:26
  • @MAS The proof is fine; the question title is all math, and should have some non-math words: http://meta.math.stackexchange.com/questions/9687/guidelines-for-good-use-of-latex-in-question-titles/9730#9730 Also compare this question to http://math.stackexchange.com/questions/212149/proof-that-overlinea-cap-b-subseteq-overlinea-cap-overlineb?rq=1 – David K Dec 31 '14 at 14:00
  • Also note that it doesn't matter that you are intersecting only two sets. – Adam Bartoš Dec 31 '14 at 14:20
  • It would seem to me that the more demanding proof would be that the closure is unique, which is the assumption you make when you assign such a property as a function, which is the core of your proof, so in my opinion, this proof is correct if you can prove the assumptions that go into it, with uniqueness being the most significant one to me. – DanielV Dec 31 '14 at 14:21
  • Possible dupe? http://math.stackexchange.com/questions/212149/proof-that-overlinea-cap-b-subseteq-overlinea-cap-overlineb?rq=1 – DanielV Dec 31 '14 at 14:24
  • If this is for a basic course, you might want to justify the first implication by appealing to the definition of closure. – ashman Dec 31 '14 at 19:39

1 Answers1

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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.

user2345215
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