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Math Help - Prove that two sets have the same cardinality

  1. #1
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    Prove that two sets have the same cardinality

    Hi there,

    I have a question to solve: if a<b and c<d for , where a,b,c and d are all rationals, how do I prove that [a,b] and [c,d] have the same cardinality.

    I know that we're supposed to show bijection between the sets, but I don't know how I would start. Any help please?
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  2. #2
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    Can you show that both sets have the same cardinality as [0,1]?
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  3. #3
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    See what the function f(x)=\frac{d-c}{b-a}(x-a)+c does for you.
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  4. #4
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    I got it, I think. I used Plato's function and it did the trick. Defunkt, your suggestion works too, but it might double the work if I used my current approach.

    Plato, question for you: how did you get that function? i
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  5. #5
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    Quote Originally Posted by dgmath View Post
    Plato, question for you: how did you get that function? i
    Fifty years of experience.
    Moreover, any linear function is a bijection on the reals.
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  6. #6
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    Plato, let me rephrase the question. What does that function signify? Is it some sort of a relation between the two sets [a,b] and [c,d] ?
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  7. #7
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    He answered that: "any linear function is a bijection on the reals."

    And, of course, he chose his linear function so that a is mapped to c and b is mapped to d.
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