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Math Help - Nagging question

  1. #1
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    Nagging question

    Let A and B be nonempty finite sets. For f:A \rightarrow B to be a function, it must satisfy this condition:

    x\in A \Rightarrow (x, f(x)) \in f.

    Now, if I make A=\emptyset and B=\emptyset, the statement of implication becomes

    x\notin A \Rightarrow (x,f(x)) \notin f, from which we know that the statement of implication is true, since

    False \Rightarrow False is a true statement. In this regard, I have a question:

    Is f:\emptyset \rightarrow \emptyset a function?
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  2. #2
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    Have you forgotten that x\notin\emptyset is a true statement?
    So x\in\emptyset is false.
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  3. #3
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    Quote Originally Posted by Plato View Post
    Have you forgotten that x\notin\emptyset is a true statement?
    So x\in\emptyset is false.
    In deed, I forgot about it. Oh, Sir, I had a brain fade at noon right after eating a full meal.
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  4. #4
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    So we see:

    0 is a function from 0 into 0

    By the way, another cute one is this:

    [EDITED OUT; CORRECTED BELOW]
    Last edited by MoeBlee; May 2nd 2010 at 12:34 PM.
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  5. #5
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    Quote Originally Posted by MoeBlee View Post
    So we see:

    0 is a function from 0 into 0

    By the way, another cute one is this:

    {1} is a function

    {1} = {{0}} = {{0} {0 0}} = {<0 0>} is a function
    No comprehende, amigo!
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  6. #6
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    Quote Originally Posted by novice View Post
    No comprehende
    That's because I made a mistake.

    I meant to say {1} is an ordered pair and {{1}} is a function.

    {{1}} = {{{0}}} = {{{0} {0 0}}} = {<0 0>} is a function
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  7. #7
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    Quote Originally Posted by MoeBlee View Post
    That's because I made a mistake.

    I meant to say {1} is an ordered pair and {{1}} is a function.

    {{1}} = {{{0}}} = {{{0}, {0, 0}}} = {<0, 0>} is a function
    MoeBlee
    I am curious at why you do not use commas in ordered pairs as is the standard notation.
    Why do you use <0 ~0> in place of the standard (0,0)?
    You know that can confuse many who use resource.

    Again, why donít you learn to use LaTeX?
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  8. #8
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    I find that unneeded commas are usually visual noise.

    The notation

    <0 0>

    is quite clear.

    I've corresponded with numerous professional and amateur mathematicians and have never had any problem with

    <0 0>

    being understood as the ordered pair with 0 as first coordinate and 0 as second coordinate.

    /

    I've already responded in another thread regarding LaTex.
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  9. #9
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    Quote Originally Posted by MoeBlee View Post
    That's because I made a mistake.

    I meant to say {1} is an ordered pair and {{1}} is a function.

    {{1}} = {{{0}}} = {{{0} {0 0}}} = {<0 0>} is a function
    I must thank you for your willingness in helping me with my learning, though I have not learned enough mathematics to understand what is being imparted to me. Hopefully, the day will come when I can recognize it and appreciate it. Mucho Gracias.
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  10. #10
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    Quote Originally Posted by novice View Post
    I have not learned enough mathematics to understand what is being imparted to me.
    It's just a kind of fun fact that derives from our particular set theoretic definition of the ordered pair operation:

    The standard definition of the ordered pair operation:

    <x y> = {{x} {x y}}

    Also, we have a "redundancy property":

    {x x} = {x}

    Also, we have the von Neumann approach to natural numbers, in which 1 = {0}.

    So:

    {{1}} = {{{0}}} = {{{0} {0}}} = {{{0} {0 0}}} = {<0 0>}

    And {<0 0>} is a function since it satisfies the set theoretic definition of a function:

    f is a function iff (f is a set of ordered pairs & there are no members <x y> and <x z> in f unless y=z)
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  11. #11
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    Quote Originally Posted by MoeBlee View Post
    I find that unneeded commas are usually visual noise.

    The notation

    <0 0>

    is quite clear.

    I've corresponded with numerous professional and amateur mathematicians and have never had any problem with

    <0 0>

    being understood as the ordered pair with 0 as first coordinate and 0 as second coordinate.

    /

    I've already responded in another thread regarding LaTex.
    That may well be the case. Nevertheless, your notation is not standard notation and has the potential to create confusion. I support Plato's remarks and request that you follow them when making future posts.
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  12. #12
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    P.S. Actually, the notation

    (x, y)

    is more ambiguous than

    <x y>.

    (x, y)

    could be taken as the ordered pair of x and y, or as an open real interval.

    But

    <x y>

    is quite clearly the ordered pair of x and y.
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  13. #13
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    Quote Originally Posted by MoeBlee View Post
    P.S. Actually, the notation

    (x, y)

    is more ambiguous than

    <x y>.

    (x, y)

    could be taken as the ordered pair of x and y, or as an open real interval.

    But

    <x y>

    is quite clearly the ordered pair of x and y.
    This thread is taking a direction beyond the question asked by the OP.

    Thread closed.
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