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Math Help - bijection help

  1. #1
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    easy set theory

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    Last edited by yellow4321; March 6th 2008 at 05:06 AM.
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  2. #2
    Lord of certain Rings
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    Quote Originally Posted by yellow4321 View Post
    how could i show a bijection between the natural numbers and set of even natural numbers 2n?also from the N and Zintergers
    Define f:\cal{N} \to \cal{Z} as
    ..........n/2 ----------when n is even
    f(n) =
    ..........(1-n)/2 ---------when n is odd

    The first one is a simple exercise to show injectivity and surjectivity. Do it for this too.

    P.S: How do we put the huge bracket for such piecewise defined function in TeX? Thanks
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  3. #3
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    \begin{array}{l}<br />
 \Phi :Z \mapsto N \\ <br />
 \Phi (z) = \left\{ {\begin{array}{*{20}c}<br />
   {2\left| z \right|} & {z \le 0}  \\<br />
   {2z - 1} & {z > 0}  \\<br />
\end{array}} \right. \\ <br />
 \end{array}
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by yellow4321 View Post
    how could i show a bijection between the natural numbers and set of even natural numbers 2n?
    I'll start you off.

    Let f: \mathbb{N} \to \mathbb{N}_{\mbox{even}} be defined by f(n) = 2n for all n \in \mathbb{N}.

    We wish to show that f is a bijective function. To show this we need to show:

    (1) f is one-to-one.

    Definition: a function f is one-to-one if and only if f(a) = f(b) implies a = b

    (2) f is onto.

    Definition: a function f is onto if every element in the range is an image of some element in the domain, that is, f: A \to B is onto if and only if for every b \in B there is an a \in A such that b = f(a).

    Now continue
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  5. #5
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    Last edited by yellow4321; March 6th 2008 at 05:07 AM.
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