Thread: Prove or disprove using boundary points

1. Prove or disprove using boundary points

Let $\displaystyle A$ and $\displaystyle C$ be subsets of $\displaystyle R^n$ with boundaries $\displaystyle B(A)$, $\displaystyle B(C)$ respectively.
Prove or disprove:
1. $\displaystyle B(A\cup C)=B(A)\cup B(C)$
2. $\displaystyle B(A\cap C)=B(A)\cap B(C)$

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

2. Originally Posted by Runty
Let $\displaystyle A$ and $\displaystyle C$ be subsets of $\displaystyle R^n$ with boundaries $\displaystyle B(A)$, $\displaystyle B(C)$ respectively.
Prove or disprove:
1. $\displaystyle B(A\cup C)=B(A)\cup B(C)$
2. $\displaystyle B(A\cap C)=B(A)\cap B(C)$
For #1. In $\displaystyle \mathbb{R}^1$ let $\displaystyle A=[0,1]~\&~C=[1,2]$.

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

3. 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.

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

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

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

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

#2: $\displaystyle B(A\cap C)=\emptyset$ and $\displaystyle B(A)\cap B(C)=\{1\}$, 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.