1. ## one prove question

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

2. Originally Posted by wopashui
Let A and C be subset of $R^n$ with boundaries B(A), B(C) respectively. Prove or disprove:
(1) B(AUC) = B(A)UB(C)
(2) B(A $\cap$C) = B(A) $\cap$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 $\,\mathbb{R}$,

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

so

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

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

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

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

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

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

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 $\,\mathbb{R}$,

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

so

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

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

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

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

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

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

4. Originally Posted by wopashui
is this situation only in $R^1$ or applied in $R^n$
The edit was written just for $\mathbb{R}$, 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.