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Thread: bijective maps

  1. #1
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    bijective maps

    Let $\displaystyle n \in \mathbb{Z^+}, X$ is nonempty.
    (i) Find a bijective map $\displaystyle f: X^\omega \times X^\omega \rightarrow X^\omega$
    (ii) If $\displaystyle A \subset B$ find an injective map $\displaystyle gA^\omega)^n \rightarrow B^\omega$

    The cartesian product $\displaystyle A_1 \times A_2 \times ... $ is the set of all $\displaystyle \omega-$tuples of elements of $\displaystyle X$ is denoted by $\displaystyle X^\omega$.

    I've tried many examples but none works... Can I get some help with these two please?
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  2. #2
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    (i) define $\displaystyle f$ for example like this: $\displaystyle f(\left<g,h\right>)=u$ where $\displaystyle u(n) = \begin{cases} g(m) & \mbox{if } n=2m \\ h(m) & \mbox{if } n=2m+1 \end{cases}$

    (ii) if $\displaystyle A$ is empty then $\displaystyle g = \emptyset$ (the empty mapping) is injective. if $\displaystyle A$ is nonempty, you can define g similarly as in (i):

    $\displaystyle g(\left<h_0, \ldots, h_{n-1}\right>) = v$ where $\displaystyle v(i) = \begin{cases} h_0(m) & \mbox{if } i=nm \\ h_1(m) & \mbox{if } i=nm+1 \\ \cdots \\ h_{n-1}(m) & \mbox{if } i=nm+n-1\end{cases}$
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  3. #3
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    Quote Originally Posted by dori1123 View Post
    Let $\displaystyle n \in \mathbb{Z^+}, X$ is nonempty.
    (i) Find a bijective map $\displaystyle f: X^\omega \times X^\omega \rightarrow X^\omega$
    This makes no sense. Did you mean $\displaystyle X\subset Z^+$ rather than $\displaystyle n \in Z^+$?

    (ii) If $\displaystyle A \subset B$ find an injective map $\displaystyle gA^\omega)^n \rightarrow B^\omega$

    The cartesian product $\displaystyle A_1 \times A_2 \times ... $ is the set of all $\displaystyle \omega-$tuples of elements of $\displaystyle X$ is denoted by $\displaystyle X^\omega$.

    I've tried many examples but none works... Can I get some help with these two please?
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    This makes no sense. Did you mean $\displaystyle X\subset Z^+$ rather than $\displaystyle n \in Z^+$?
    Hi,
    what exactly doesn't make sense? condition $\displaystyle n \in \mathbb{Z}^+$ applies to part (ii).
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  5. #5
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    it is $\displaystyle n \in \mathbb{Z^+}$, it's a problem from the book Topology by Munkres
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  6. #6
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    When you write $\displaystyle f(\left<g,h\right>) = ...$, is this $\displaystyle \left<g,h\right>$ the cyclic group generated by $\displaystyle g$ and $\displaystyle h$?
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  7. #7
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    no, it is an ordered pair.
    $\displaystyle f: X^\omega \times X^\omega \rightarrow X^\omega$ means that $\displaystyle f$ takes a pair of functions from $\displaystyle \omega$ to $\displaystyle X$ and returns another function from $\displaystyle \omega$ to $\displaystyle X$.
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  8. #8
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    Thanks I just want to make sure because I've been using the symbol < > for cyclic groups..

    Can you also help me with one more problem?
    with the same conditions given,
    find a bijective map $\displaystyle f: X^n \times X^\omega \rightarrow X^\omega$
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  9. #9
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    if you understand previous two solutions, this one should be no problem for you..
    define $\displaystyle f(\left<g,h\right>)=u$ where $\displaystyle u(i) = \begin{cases} g(i) & \mbox{if } i<n \\ h(i-n) & \mbox{if } i \ge n \end{cases}$

    can you verify it is well-defined, injective and surjective?
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