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Math Help - set theory problem, open and close intervals equivalance

  1. #1
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    Exclamation set theory problem, open and close intervals equivalance

    hello there,
    i am trying to prove a set theory problem relating open and half open intervals. help needed.
    1- [0,1] ~ (a,b)
    2- [0,1] ~ [a,b)
    3- [0,1] ~ (a,b]
    where a,b belongs to R and ~ is equivalent sign.

    for question 2, [0,1] ~ [a,b)
    i have defined a function such that f(x) = (bx+a/2) / 2 for all x belongs to [0,1]. is that correct??
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  2. #2
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    Re: set theory problem, open and close intervals equivalance

    Two sets are "equivalent" if and only if there exist an invertible function that maps every point in the first set to every point in the other set. Your first problem requires that you find an invertible function that maps (0, 1) to (a, b). You want to use f(x)= (bx+ a/2)/2 which is invertible. But f(0)= a/4 and f(1)= b/2+ a/4. That does NOT map (0, 1) to (a, b).

    But there is a more serious problem. To do this problem you should know that a continuous invertible function must map open sets to open sets and closed sets to closed sets. Since [0, 1] is closed but none of (a, b), [a, b), or (a, b] are closed, no continuous, and certainly no linear function will work. I would recommend something like: map every irrational number to itself and then use the fact that the set of rational numbers in both [0, 1], (a, b), [a, b), (a, b] are countable.
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  3. #3
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    Re: set theory problem, open and close intervals equivalance

    Quote Originally Posted by MathGeni04 View Post
    where a,b belongs to R and ~ is equivalent sign.
    The name of the sign is irrelevant; it's what it denotes that matters.

    See also this thread.
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    Re: set theory problem, open and close intervals equivalance

    Quote Originally Posted by HallsofIvy View Post
    Two sets are "equivalent" if and only if there exist an invertible function that maps every point in the first set to every point in the other set. Your first problem requires that you find an invertible function that maps (0, 1) to (a, b). You want to use f(x)= (bx+ a/2)/2 which is invertible. But f(0)= a/4 and f(1)= b/2+ a/4. That does NOT map (0, 1) to (a, b).

    But there is a more serious problem. To do this problem you should know that a continuous invertible function must map open sets to open sets and closed sets to closed sets. Since [0, 1] is closed but none of (a, b), [a, b), or (a, b] are closed, no continuous, and certainly no linear function will work. I would recommend something like: map every irrational number to itself and then use the fact that the set of rational numbers in both [0, 1], (a, b), [a, b), (a, b] are countable.
    would you mind giving any examples of f(x) which can be used in these problem. i am new to this stuff, dnt have much grip. i'll much obliged.
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    Re: set theory problem, open and close intervals equivalance

    Quote Originally Posted by MathGeni04 View Post
    would you mind giving any examples of f(x) which can be used in these problem.
    Have you seen the tread I linked to? Do you think I kept in my bookmarks for 1.5 years in vain?
    Thanks from MathGeni04 and topsquark
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  6. #6
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    Re: set theory problem, open and close intervals equivalance

    Quote Originally Posted by emakarov View Post
    Have you seen the tread I linked to? Do you think I kept in my bookmarks for 1.5 years in vain?
    i hv seen it. thnx
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  7. #7
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    Re: set theory problem, open and close intervals equivalance

    Quote Originally Posted by MathGeni04 View Post
    i am trying to prove a set theory problem relating open and half open intervals. help needed.
    1- [0,1] ~ (a,b)
    2- [0,1] ~ [a,b)
    3- [0,1] ~ (a,b]
    I see that you have solved this question. Here is some advice for future questions like this.
    You can show that [0,1]\sim (0,1]\sim [0,1)\sim (0,1).
    We do that by point shifting of a countable set. Define a sequence f_0=0,~f_n=n^{-1},~n\in\mathbb{Z}^+.

    Now if x\in [0,1] and if (\exists k)[x=f_k] then define \pi:~x\mapsto f_{k+1} or else \pi:~x\mapsto x.
    That shows that [0,1]\overset{\pi}\sim(0.1]. Similar functions made with minor adjustments in \pi.

    Note that if a<b then \sigma: x\mapsto a+(b-a)x makes [0,1]\overset{\sigma}\sim [a,b].
    Thanks from topsquark
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