# $\overline{X\cap Y} \subseteq \overline{X} \cap \overline{Y}$

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?

• It is a quite good proof. Dec 31, 2014 at 12:58
• It seems correct yes
– user169373
Dec 31, 2014 at 12:58
• 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? Dec 31, 2014 at 13:04
• 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? Dec 31, 2014 at 13:04
• @abiessu Here overline indicates closure. Dec 31, 2014 at 13:06