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Math Help - Functions on the empty set

  1. #16
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    After rereading this thread, I can see wisdom in that position.
    Donít you think that these are ridiculous arguments?
    Why is it ridiculous to try define something which is clearly a pretty big requirement in other areas of math? May not be very interesting cases, but it exhausts them, something (I believe) is an important foundation of mathematics.
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  2. #17
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    "Anyway, and going back to the OP question: if A\,\,\,or\,\,\,B=\emptyset , then A\times B=\emptyset and thus there's one unique function from or to the empty set."

    @Tonio - Sorry but this is not making sense to me. As a funtion F, has to be a subset of AXB (which itself is empty) - so doesn't it mean it can't have a subset? Then in other words no such function exists. Maybe I'm not understanding it - is there a example you can give?


    Quote Originally Posted by tonio View Post
    It would really be interesting to know what grad school did Plato attend since they obviously do not abide by ZFC and usual set theory rules. Not taking into consideration the empty set as a set (if this is what Plato really meant) has rather far-reaching consequences in some definite areas in maths (for example, in combinatorics).

    Anyway, and going back to the OP question: if A\,\,\,or\,\,\,B=\emptyset , then A\times B=\emptyset and thus there's one unique function from or to the empty set.

    As for 0^0=1 : it certainly isn't defined but x^x\xrightarrow [x\to 0^+]{}1 , so one can safely define 0^0=1 as long as we restrict ourselves to positive reals.


    Tonio
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  3. #18
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    Quote Originally Posted by aman_cc View Post
    "Anyway, and going back to the OP question: if A\,\,\,or\,\,\,B=\emptyset , then A\times B=\emptyset and thus there's one unique function from or to the empty set."

    @Tonio - Sorry but this is not making sense to me. As a funtion F, has to be a subset of AXB (which itself is empty) - so doesn't it mean it can't have a subset? Then in other words no such function exists. Maybe I'm not understanding it - is there a example you can give?
    The set itself is always a subset of itself. I don't think the empty set is any exception.
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  4. #19
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    Quote Originally Posted by bleys View Post
    The set itself is always a subset of itself. I don't think the empty set is any exception.

    Sorry for my mistake. Yes - but I'm still not able to get F into my head. What will F be like?
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  5. #20
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    Quote Originally Posted by tonio View Post
    It would really be interesting to know what grad school did Plato attend since they obviously do not abide by ZFC and usual set theory rules.Tonio
    He obviously studied under Socrates!
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  6. #21
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    You mean an explicit representation? Hmm, I don't think there is one. It's the function that sends nothing to nothing; it only exists out of how logic rules are defined, I suppose.
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  7. #22
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    Quote Originally Posted by HallsofIvy View Post
    He obviously studied under Socrates!

    Hehe...good one. Let's hope that one of his/her students won't be like Aristotle scientificwise, since this last one got most of his "scientific" deductions wrong...
    Anyway, Aristotle kicked ass as philosopher and logician.

    Tonio
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  8. #23
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    Quote Originally Posted by aman_cc View Post
    Sorry for my mistake. Yes - but I'm still not able to get F into my head. What will F be like?
    @aman_cc: Consider a function from A=\{a\} to any B, let's say B=\mathbb{N} for more convenience.
    f is defined by f(a), since a is the only element of A,\ f has the form \{(a,n)\} for some n\in\mathbb{N}.

    With A empty, well there is no element which needs an image, therefore no pair (x,f(x)) in f, i.e. f=\emptyset.

    Of course it's a pathological case, but you can check it satisfies the definition of a function from A to B: \forall x\in A\ \exists !y\in B\ ((x,y)\in f)
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