one prove question

**wopashui** · September 14th 2010, 05:23 PM

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)

**undefined** · September 14th 2010, 05:33 PM

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

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

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

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

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

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

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

**wopashui** · September 19th 2010, 11:20 AM

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 $R^n$

**undefined** · September 19th 2010, 11:32 AM

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.

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