# Union of sets in R^n

• January 11th 2012, 04:15 PM
I-Think
Union of sets in R^n
The question states:
For $A,B\subset{{R^n}}$, prove that $\overline{A\cup{B}}=\overline{A}\cup{\overline{B}}$

Shouldn't it be $\overline{A}\cap{\overline{B}}$ by De Morgan's Laws?
Also, I thought of this counter example in ${R{^2}}$

Let $A,B\subset{{R{^2}}}$
$x\in{A}$ if $x_1$ even, $x_2=3k, k\in{\mathbb{Z}}$
$x\in{B}$ if $x_1$ even, $x_2=4k, k\in{\mathbb{Z}}$

Now consider $y=(4,8)$. $y\in{\overline{A}}$, thus $y\in{\overline{A}\cup{\overline{B}}}$.
But $y$ not in $\overline{A\cup{B}}$
Thus $\overline{A\cup{B}}\neq{\overline{A}\cup{\overline {B}}}$

Where is my mistake or is this the question's mistake?
• January 11th 2012, 04:48 PM
Plato
Re: Union of sets in R^n
Quote:

Originally Posted by I-Think
The question states:
For $A,B\subset{{R^n}}$, prove that $\overline{A\cup{B}}=\overline{A}\cup{\overline{B}}$

Shouldn't it be $\overline{A}\cap{\overline{B}}$ by De Morgan's Laws?

I think you are miss-reading the question.
$\overline{A}$ is not the complement of $A$. It is the closure of $A.$

The theorem is: The closure of a union is the union of closures.