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Math Help - Equinumerous

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    Equinumerous

    Show that the following pairs of sets S and T are equinumerous by finding a specific bijection between the two sets in each pair.

    a. S= [0,1] and T = [1,3]

    b. S= [0,1] and T = [0, infinity]
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    Quote Originally Posted by GoldendoodleMom View Post
    Show that the following pairs of sets S and T are equinumerous by finding a specific bijection between the two sets in each pair.

    a. S= [0,1] and T = [1,3]
    Think geometrically. Length of [0,1] is 1 and length of [1,3] is 2. Thus, strech [0,1] to [0,2]. Now shift this interval by one to the right this gives [1,3].

    Now the streching can be desribed by function f :[0,1] \to [0,2] by f(x) = 2x. And the shifting can be described by g:[0,2]\to [1,3] by g(x) = x+1. Thus, g\circ f: [0,1] \to [1,3] is what you are looking for. Finally g(f(x)) = 2x+1 is the mapping you want.
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  3. #3
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    Quote Originally Posted by GoldendoodleMom View Post
    Show that the following pairs of sets S and T are equinumerous by finding a specific bijection between the two sets in each pair.

    b. S= [0,1] and T = [0, infinity]
    Consider f:[0,1) \to [0, \infty) defined by f(x)=\frac{x}{1-x} for x \in [0,1)

    RonL
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    Senior Member bkarpuz's Avatar
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    Exclamation

    Quote Originally Posted by CaptainBlack View Post
    Consider f:[0,1) \to [0, \infty) defined by f(x)=\frac{x}{1-x} for x \in [0,1)

    RonL
    Also an alternative is to use \tan and/or \tan^{-1} (inverse of \tan), these really help too much.
    f:[0,1)\to[0,\infty)
    ......... t\to f(t)=\tan\bigg(\frac{\pi}{2}t\bigg)
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