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Thread: some play on empty sets

  1. #1
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    some play on empty sets

    Hello

    I just have some general question regarding functions. Consider the sets A and B. Now if $\displaystyle A\neq \varnothing\mbox{ and }B=\varnothing$, then the definition of the function$\displaystyle f:A\to B$ fails but there is still a function $\displaystyle g:B\to A$, even though in both cases $\displaystyle A\times B=\varnothing$ . So if we define $\displaystyle ^AB$ as the set of functions from A to B, then if
    $\displaystyle B=\varnothing\mbox{ and } A\neq \varnothing$, then

    $\displaystyle ^AB=\varnothing\mbox{ and } ^BA=\{\varnothing\}$

    is that right reasoning ? I need this result in some proof.
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  2. #2
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    Re: some play on empty sets

    Quote Originally Posted by issacnewton View Post
    I just have some general question regarding functions. Consider the sets A and B. Now if $\displaystyle A\neq \varnothing\mbox{ and }B=\varnothing$, then the definition of the function$\displaystyle f:A\to B$ fails but there is still a function $\displaystyle g:B\to A$, even though in both cases $\displaystyle A\times B=\varnothing$ . So if we define $\displaystyle ^AB$ as the set of functions from A to B, then if $\displaystyle B=\varnothing\mbox{ and } A\neq \varnothing$, then
    $\displaystyle ^AB=\varnothing\mbox{ and } ^BA=\{\varnothing\}$
    is that right reasoning ? I need this result in some proof.
    This question can start an argument. The emptyset is problematic.
    That said, I will give a quote directly from Halmos.
    If each of $\displaystyle X~\&~Y$ is a set and $\displaystyle Y^X$ is the set of all functions $\displaystyle X\to Y$ then the quote
    "$\displaystyle (i) Y^{\emptyset}$ has exactly one element, namely $\displaystyle \emptyset$, whether $\displaystyle Y$ is empty or not, and (ii) if $\displaystyle X$ is not empty, then $\displaystyle \emptyset^X$ is empty."
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  3. #3
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    Re: some play on empty sets

    Plato, thats what I am saying too, am I not ?
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    Re: some play on empty sets

    Quote Originally Posted by issacnewton View Post
    Plato, thats what I am saying too, am I not ?
    I think so. But that notation is confusing.
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  5. #5
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    Re: some play on empty sets

    Thats what Daniel (Velleman) uses in his book. Just following him.......
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