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Math Help - If S has the same number of elements as T, then T has the same number as S...

  1. #1
    No one in Particular VonNemo19's Avatar
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    If S has the same number of elements as T, then T has the same number as S...

    Hi. The proof that I'm working is in the title. The Defition of "same number as" is given as follows: (1) Two sets S and T are said to have the same nuber of elements if each element of S can be paired with precisely one element of  T in such a way that every element of  T is paired with precisely one element of S.

    Notation: If \pi is a pairing of the elements of S with with those of Tand the element sof S is paired in \pi to the element t of T , we shall write s\overbrace{\leftrightarrow}^{\pi}{t} (I don't know how to put the \pi above the \leftrightarrow without the \overbrace).
    I don't really understand the book's proof: Since S has the same number of elements as T, we can select a pairing between the elements of S and T in accordance with (1). We define a pairing as follows: If  s is paired with t by the selected pairing \pi, then pair  t with s. That is if s\overbrace{\leftrightarrow}^{\pi}{t}, then  t is paired with s to form the pairing of the elements T with those of S. If the original pairing satisfied (1), then so will the new pairing. Specifically, since  \pi had each element  T paired with a unique element of S, then the second pairing also has this property. Therefore, T has the same number of elements as  S.

    Please help me to understand why this proposition is not trivial, and also the procedure of the proof.
    Last edited by VonNemo19; July 21st 2012 at 10:04 AM.
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    Re: If S has the same number of elements as T, then T has the same number as S...

    Quote Originally Posted by VonNemo19 View Post
    Hi. The proof that I'm working is in the title. The Defition of "same number as" is given as follows: (1) Two sets S and T are said to have the same number of elements if each element of S can be paired with precisely one element of  T in such a way that every element of  T is paired with precisely one element of S.
    Notation: If \pi is a pairing of the elements of S with with those of Tand the element sof S is paired in \pi to the element t of T , we shall write s\overbrace{\leftrightarrow}^{\pi}{t} (I don't know how to put the \pi above the \leftrightarrow without the \overbrace).
    I don't really understand the book's proof: Since S has the same number of elements as T, we can select a pairing between the elements of S and T in accordance with (1). We define a pairing as follows: If  s is paired with t by the selected pairing \pi, then pair  t with s. That is if s\overbrace{\leftrightarrow}^{\pi}{t}, then  t is paired with s to form the pairing of the elements T with those of S. If the original pairing satisfied (1), then so will the new pairing. Specifically, since  \pi had each element  T paired with a unique element of S, then the second pairing also has this property. Therefore, T has the same number of elements as  S. Please help me to understand why this proposition is not trivial, and also the procedure of the proof.
    I agree it is trivial from my point of view. But without seeing your notes, I cannot sure of that.
    If S has the same same number of elements as T there is a bijection \pi:S\to T.
    But \pi^{-1}:T\to S is also a bijection.
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    Re: If S has the same number of elements as T, then T has the same number as S...

    The crucial part that a bijection is invertible. If f is a bijection from A to B, then its inverse is a bijection from B to A.
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