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Math Help - cardinal numbers

  1. #1
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    cardinal numbers

    Help, I'm drowning!! In proofs. I am sooooo lost in this class.

    Questions:
    1. Prove that the set of positive real numbers has cardinal number c. (Hint: try the map x --> e^x.

    2. Prove that the open unit interval 9the set of all real x with 0 < x < 1) has cardinal number c. (Hint: Use ex. 1 and the map x --> x/(1 + x), defined on all positive real numbers).

    3. Whatis the cardinal number of the closed unit interval (all x with 0 < x < 1)? Of the half open interval (all xx with0 < x < 1)?
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  2. #2
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    Quote Originally Posted by gatordeb View Post
    Help, I'm drowning!! In proofs. I am sooooo lost in this class.

    Questions:
    1. Prove that the set of positive real numbers has cardinal number c. (Hint: try the map x --> e^x.

    2. Prove that the open unit interval 9the set of all real x with 0 < x < 1) has cardinal number c. (Hint: Use ex. 1 and the map x --> x/(1 + x), defined on all positive real numbers).

    3. Whatis the cardinal number of the closed unit interval (all x with 0 < x < 1)? Of the half open interval (all xx with0 < x < 1)?
    If I'm understanding correctly, you are being to asked to show there exists a bijection between (1) the set of reals and the set of positive reals, (2) the set of reals and the open unit interval, (3) etc.

    Can you prove f(x) = e^x defined on the reals is a bijection?
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  3. #3
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    I don't know how. I'm telling you, I'm lost.
    If it helps, the book we're using is Kaplansky's Set Theory and Metric Spaces.
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  4. #4
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    Quote Originally Posted by gatordeb View Post
    I don't know how. I'm telling you, I'm lost.
    If it helps, the book we're using is Kaplansky's Set Theory and Metric Spaces.
    Define \begin{aligned}f:\ &\mathbb{R}\to\{x\,|\,x\in\mathbb{R}\land x>0\}\\&x\mapsto e^x\end{aligned}

    First show f is injective (one-to-one), then show f is surjective (onto).

    Intuitively/geometrically, f being injective means it passes the horizontal line test.

    f being surjective means that the codomain equals the image.

    Injective, Surjective and Bijective
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