1. ## one prove question

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

2. Originally Posted by wopashui
Let A and C be subset of $\displaystyle R^n$ with boundaries B(A), B(C) respectively. Prove or disprove:
(1) B(AUC) = B(A)UB(C)
(2) B(A$\displaystyle \cap$C) = B(A)$\displaystyle \cap$B(C)
I haven't studied this formally per se, but both (1) and (2) seem to fail when we consider the simple case of two intersecting spheres in R^3, or even just intersecting discs in R^2.

Edit: Actually easier to write in $\displaystyle \,\mathbb{R}$,

let $\displaystyle A = [0,3], C = [1,2]$,

so

$\displaystyle B(A) = \{0, 3\}$,

$\displaystyle B(C) = \{1, 2\}$,

$\displaystyle B(A \cup C) = \{0, 3\}$,

$\displaystyle B(A) \cup B(C) = \{0,1,2,3\}$,

$\displaystyle B(A \cap C) = \{1,2\}$,

$\displaystyle B(A) \cap B(C) = \emptyset$

3. Originally Posted by undefined
I haven't studied this formally per se, but both (1) and (2) seem to fail when we consider the simple case of two intersecting spheres in R^3, or even just intersecting discs in R^2.

Edit: Actually easier to write in $\displaystyle \,\mathbb{R}$,

let $\displaystyle A = [0,3], C = [1,2]$,

so

$\displaystyle B(A) = \{0, 3\}$,

$\displaystyle B(C) = \{1, 2\}$,

$\displaystyle B(A \cup C) = \{0, 3\}$,

$\displaystyle B(A) \cup B(C) = \{0,1,2,3\}$,

$\displaystyle B(A \cap C) = \{1,2\}$,

$\displaystyle B(A) \cap B(C) = \emptyset$
is this situation only in $\displaystyle R^1$ or applied in $\displaystyle R^n$

4. Originally Posted by wopashui
is this situation only in $\displaystyle R^1$ or applied in $\displaystyle R^n$
The edit was written just for $\displaystyle \mathbb{R}$, but in order to disprove a statement you only need one counterexample.

Like I said though I haven't studied this formally but it seems all right.