one prove question

• Sep 14th 2010, 05:23 PM
wopashui
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)
• Sep 14th 2010, 05:33 PM
undefined
Quote:

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\$
• Sep 19th 2010, 11:20 AM
wopashui
Quote:

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\$
• Sep 19th 2010, 11:32 AM
undefined
Quote:

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.