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Thread: Very interesting question about sets

  1. #1
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    Very interesting question about sets

    A = (a,b)
    B = [a,b)
    C = [a,b]
    D = (a,b]

    , a,b are real numbers

    Which one of these sets is the biggest?

    (Proof is not necessary, I only need to know the correct answer)
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  2. #2
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    Quote Originally Posted by Kiki View Post
    A = (a,b)
    B = [a,b)
    C = [a,b]
    D = (a,b]

    , a,b are real numbers

    Which one of these sets is the biggest?

    (Proof is not necessary, I only need to know the correct answer)

    Bad question: there's no biggest nor smallest. All of them have the same cardinality (assuming, of course, $\displaystyle a<b$...)

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Bad question: there's no biggest nor smallest. All of them have the same cardinality (assuming, of course, $\displaystyle a<b$...)

    Tonio
    Ok, thank you.
    Another question:
    I have prooved that a function converges in all these 4 sets.
    I would just like to use the "biggest" of them to show that it converges in the a larger area (like trying to include all the 4 of them, in a sentence) and I am wondering which one to use.
    Maybe the set "C" ?

    (assuming , $\displaystyle a<b$...)
    Last edited by Kiki; Apr 21st 2010 at 02:50 PM.
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  4. #4
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    Quote Originally Posted by Kiki View Post
    A = (a,b)
    B = [a,b)
    C = [a,b]
    D = (a,b]

    , a,b are real numbers
    Which one of these sets is the biggest?
    Here is a hint. Define $\displaystyle f:[0,1]\to (0,1)$ by $\displaystyle f(0)=\frac{1}{2},~ f(1)=\frac{1}{3}, f(1/n)=\frac{1}{n+1}, n\ge3,~x\text{ otherwise} $.
    It is easy to show that $\displaystyle f$ is a bijection.
    Thus we see that $\displaystyle [0,1]~\&~(0,1)$ are ‘same size’.
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  5. #5
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    Quote Originally Posted by Kiki View Post
    I have prooved that a function converges in all these 4 sets. I would just like to use the "biggest" of them to show that it converges in the a larger area (like trying to include all the 4 of them, in a sentence) and I am wondering which one to use.
    Use set $\displaystyle C$ because each of the others are subsets.
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  6. #6
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    Quote Originally Posted by Plato View Post
    Use set $\displaystyle C$ because each of the others are subsets.
    Ok thank you, my question is answered now.
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  7. #7
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    But it sounds weird to me that A,B,D are subsets of C, but C is not "bigger" than them.

    (My major is not mathematics so I havent studied pure topology)
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  8. #8
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    Quote Originally Posted by Kiki View Post
    But it sounds weird to me that A,B,D are subsets of C, but C is not "bigger" than them.
    In this case what matters is being the supper-set.
    That is, if I understand what you mean by “a function converges”.
    For example: If $\displaystyle f$ is continuous on $\displaystyle [a,b]$ then it is continuous on $\displaystyle (a,b)$.
    "Bigger than" has not meaning in this context.
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  9. #9
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    Quote Originally Posted by Plato View Post
    what you mean by “a function converges”.
    btw I wanted to say "a series of function converges", sorry

    Quote Originally Posted by Plato View Post
    In this case what matters is being the supper-set.
    That is, if I understand what you mean by “a function converges”.
    For example: If $\displaystyle f$ is continuous on $\displaystyle [a,b]$ then it is continuous on $\displaystyle (a,b)$.
    "Bigger than" has not meaning in this context.
    Now its pretty clear, thank you for helping me
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