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Math Help - bijective proof

  1. #1
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    bijective proof

    Hi everyone, I am stuck on the following question, can anyone help!

    let A be a finite set with m elements (m is a member of N). And suppose x is an object that is not a member of A
    Prove that A U {x} has m+1 elemetns.

    The answer is soo obvious but just dont know how to prove! Thanks so much!
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  2. #2
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    Quote Originally Posted by withoutaclue View Post
    Hi everyone, I am stuck on the following question, can anyone help!

    let A be a finite set with m elements (m is a member of N). And suppose x is an object that is not a member of A
    Prove that A U {x} has m+1 elemetns.

    The answer is soo obvious but just dont know how to prove! Thanks so much!
    Hint: Define a one-one onto mapping between A U {x} and set {1,2,3,....,m,m+1}
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  3. #3
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    Quote Originally Posted by withoutaclue View Post
    let A be a finite set with m elements (m is a member of N). And suppose x is an object that is not a member of A
    Prove that A U {x} has m+1 elements.
    By definition there is a bijection f:A \leftrightarrow \left\{ {1,2, \cdots ,m} \right\}.
    Define g:A \cup \{ x\}\mapsto \{1,2,\cdots,m,m+1\} \;,\;g(y) = \left\{ {\begin{array}{rl}<br />
   {f(y),} & {y \ne x}  \\   {m + 1,} & {y = x}  \\ \end{array} } \right.

    Prove that g is a bijection.
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  4. #4
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    thanks very much
    it gave me some clue now
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