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Math Help - Metric spaces distances... Help

  1. #1
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    Metric spaces distances... Help

    Could someone please check my answers, especially if my proves are enough to answer the question.

    True or False.. If true prove it or else give a counter example

    (a) dist (x; {x}) = 0.
    True. Since the inf(x,{x})=0.

    (b) x ∈ A ⇒ dist (x;A) = 0.
    True. Since x ∈ A, the inf(x,y|y∈A)=0.

    (c) dist (x;A) = 0 ⇒ x ∈ A.
    False. eg. x=5 ; A=(5,6]

    (d) For any subsets A; B; C of a metric space X,
    dist (A;B) ≤ dist (A;C) + dist (C;B).
    False. eg. A=[0,2], B=[4,6], C=[1,5]

    (e) A ∩ B ̸= ∅ ⇒ dist (A;B) = 0
    True. Since there exists an x ∈ A and x ∈ B then the inf(x,x)=0

    (f) dist (A;B) = 0 ⇒ A ∩ B ̸= ∅
    False. A=[0,2] B=(2,3)

    (g) A ⊂ B ⇒ dist (B;C) ≤ dist (A;C)
    True. I can visualize it, but cant proof this 1 formally
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  2. #2
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    Quote Originally Posted by Dreamer78692 View Post
    True or False.. If true prove it or else give a counter example
    (g) A ⊂ B ⇒ dist (B;C) ≤ dist (A;C)
    True. I can visualize it, but cant proof this 1 formally
    If N\subseteq M\subseteq \mathbb{R} then is it true \inf(M)\le\inf(N)~?

    If x\in C~\&~a\in A is it true that is it true that d(a,x)\in \{d(y,x):y\in B\}~?
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  3. #3
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    So since the inf(B)≤ inf(A)
    then dist (B;C) ≤ dist (A;C)

    Are the other answers right???
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  4. #4
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    Quote Originally Posted by Dreamer78692 View Post
    So since the inf(B)≤ inf(A)
    then dist (B;C) ≤ dist (A;C)
    Are the other answers right???
    Not quite. But the idea is correct.
    From \{d(x,y):x\in A~\&~y\in C\}\subseteq\{d(z,w):z\in B~\&~w\in C\} that follows.

    The others seem correct.
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