Prove or disprove using boundary points

**Runty** · September 15th 2010, 08:50 AM

Let $A$ and $C$ be subsets of $R^n$ with boundaries $B(A)$ , $B(C)$ respectively.
Prove or disprove:

$B(A\cup C)=B(A)\cup B(C)$
$B(A\cap C)=B(A)\cap B(C)$

I haven't gotten to look much at boundaries yet, so I could use a hand.

**Plato** · September 15th 2010, 10:52 AM

Originally Posted by Runty

Let $A$ and $C$ be subsets of $R^n$ with boundaries $B(A)$ , $B(C)$ respectively.
Prove or disprove:

$B(A\cup C)=B(A)\cup B(C)$
$B(A\cap C)=B(A)\cap B(C)$

For #1. In $\mathbb{R}^1$ let $A=[0,1]~\&~C=[1,2]$ .

For #2. In $\mathbb{R}^1$ let $A=[0,1)~\&~C=(1,2]$ .

**Runty** · September 16th 2010, 11:19 AM

I'm afraid that doesn't help that much, or at least I don't know how to use that information. The source material we were given relating to boundaries is very scant on details and has no practice problems to speak of.

The question itself, frankly, is not very clear on what it's looking for.

**Plato** · September 16th 2010, 11:26 AM

Originally Posted by Runty

I'm afraid that doesn't help that much, or at least I don't know how to use that information. The source material we were given relating to boundaries is very scant on details and has no practice problems to speak of.

For #1. In $\mathbb{R}^1$ let $A=[0,1]~\&~C=[1,2]$ .
$\beta(A)=\{0,1\}~,~\beta(C)=\{1,2\}~\&~\beta(A\cup C)=\{0,2\}$

For #2. In $\mathbb{R}^1$ let $A=[0,1)~\&~C=(1,2]$ .
$\beta(A)=\{0,1\}~,~\beta(C)=\{1,2\}~\&~\beta(A\cap C)=\emptyset$

**Runty** · September 17th 2010, 08:07 AM

Okay, so far I've gotten this through Plato's suggestions (which I'm assuming are meant to disprove each equality):

#1: $B(A\cup C)=\{0,2\}$ and $B(A)\cup B(C)=\{0,1,2\}$ , so it is disproven.

#2: $B(A\cap C)=\emptyset$ and $B(A)\cap B(C)=\{1\}$ , so it is disproven.

Am I on the right track, or did I make a mistake somewhere? I really can't be sure because of how little information I've been able to obtain on boundaries.

EDIT: Made a mistake with notation on second answer. I put in the symbols for union instead of intersection. Fixed.

Thread: Prove or disprove using boundary points

LinkBack

Thread Tools

Display

Prove or disprove using boundary points

Similar Math Help Forum Discussions

identify interior points, boundary points, ......

Prove a set is open iff it does not contain its boundary points

Need help understanding interior points, boundary points, isolated points, etc.

Prove or Disprove. Please help.

prove or disprove

Search Tags