- #1
SqueeSpleen
- 141
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I was trying to prove:
[itex]d(A,B) = d( \overline{A}, \overline{B} )[/itex]
I "proved" it using the following lemmas:
Lemma 1:
[itex]d(A,B) = \inf \{ d(x,B) \}_{x \in A} = \inf \{ d(A,y) \}_{y \in B}[/itex]
(By definition we have: [itex]d(A,B) = \inf \{ d(x,y) \}_{x \in A, y \in B}[/itex] )
Lemma 2:
[itex]d(x_{0},A) = d(x_{0}, \overline{A})[/itex]
Proof body:
[itex]d(A,B) \underbrace{=}_{L1} \inf \{ d(x,B) \}_{x \in A} \underbrace{=}_{L2} \inf \{ d(x, \overline{B}) \}_{x \in A} \underbrace{=}_{L1} \inf \{ d(A,y) \}_{y \in \overline{B}} \underbrace{=}_{L2} \inf \{ d(\overline{A},y) \}_{y \in \overline{B}} \underbrace{=}_{L1} d(\overline{A}, \overline{B})[/itex]
The problem is that... I have the first lemma proved in the textbook, but the second lemma isn't in it and I couldn't prove it (That's why I said "proved" instead of proved).
I know that [itex] d( x , \overline{B} ) \leq d(x,B) [/itex] almost trivially because by definition is the infimum of a set and the infimum of a subset must be equal or higher.
But all the things I though are useful to prove [itex] d( x , \overline{B} ) \leq d(x,B) [/itex] not [itex] d( x , \overline{B} ) \geq d(x,B) [/itex].
If [itex] d(x,B) = 0[/itex] I know it's true because of a theorem that implies it.
But if [itex] d(x,B) > 0[/itex] I can't prove [itex] d( x , \overline{B} ) \geq d(x,B) [/itex]
[itex]d(A,B) = d( \overline{A}, \overline{B} )[/itex]
I "proved" it using the following lemmas:
Lemma 1:
[itex]d(A,B) = \inf \{ d(x,B) \}_{x \in A} = \inf \{ d(A,y) \}_{y \in B}[/itex]
(By definition we have: [itex]d(A,B) = \inf \{ d(x,y) \}_{x \in A, y \in B}[/itex] )
Lemma 2:
[itex]d(x_{0},A) = d(x_{0}, \overline{A})[/itex]
Proof body:
[itex]d(A,B) \underbrace{=}_{L1} \inf \{ d(x,B) \}_{x \in A} \underbrace{=}_{L2} \inf \{ d(x, \overline{B}) \}_{x \in A} \underbrace{=}_{L1} \inf \{ d(A,y) \}_{y \in \overline{B}} \underbrace{=}_{L2} \inf \{ d(\overline{A},y) \}_{y \in \overline{B}} \underbrace{=}_{L1} d(\overline{A}, \overline{B})[/itex]
The problem is that... I have the first lemma proved in the textbook, but the second lemma isn't in it and I couldn't prove it (That's why I said "proved" instead of proved).
I know that [itex] d( x , \overline{B} ) \leq d(x,B) [/itex] almost trivially because by definition is the infimum of a set and the infimum of a subset must be equal or higher.
But all the things I though are useful to prove [itex] d( x , \overline{B} ) \leq d(x,B) [/itex] not [itex] d( x , \overline{B} ) \geq d(x,B) [/itex].
If [itex] d(x,B) = 0[/itex] I know it's true because of a theorem that implies it.
But if [itex] d(x,B) > 0[/itex] I can't prove [itex] d( x , \overline{B} ) \geq d(x,B) [/itex]
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