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Math Help - Functions on the empty set

  1. #1
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    Functions on the empty set

    These kind of pathologies hurt my brain... help me out:
    Formally a function F from A to B is a subset of AxB such that for any a in A there exists a unique b in B with (a,b) in F.
    How many functions are there from the empty set to the empty set? Now from the definition, "for any a in A" fails already, so everything vacuously follows. Since the only subset of the Cartesian product is the empty set, there is only one function. Is this correct? Is it actually a function or just a relation?

    What if A is empty and B is not? Following the steps again, there is only one function (relation)?

    What if A is non-empty and B is empty? Then there is no function at all? But the empty set is still a subset of the Cartesian product; is it just a relation then?
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  2. #2
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    Preface: Anything that anyone can say will be contradicted by someone.
    My graduate education was in a school where the empty set was not allowed; there are no empty point-sets.
    But I understand that is not practical in contemporary mathematics.
    That said I would simply ask you to consider these facts.
    If b\not= 0 then b^0=1.
    If b\not= 0 then 0^b=0.
    BUT 0^0 is not defined.
    Does any that help you?
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Plato View Post
    Preface: Anything that anyone can say will be contradicted by someone...

    ... I would simply ask you to consider these facts:


    a) if b\not= 0 then b^0=1.

    b) if b\not= 0 then 0^b=0.

    ... but 0^0 is not defined...
    Your preface is perfectly correct... but to contradict what You say it will be enough a trivial calculator. Please perform these steps...

    a) select among the accessories of your pc the function 'calculator'...

    b) perform this sequence of instructions : 0; x^y; 0; = ...

    c) please observe the number on the display...

    Does it help You?...

    Kind regards

    \chi \sigma
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  4. #4
    MHF Contributor undefined's Avatar
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    Here's some info on 0^0 from Wikipedia.
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  5. #5
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    So this is all a matter of convention and convenience?

    a school where the empty set was not allowed; there are no empty point-sets
    Can I ask in what areas of math this is the case? I thought the empty set is fundamental from set theory, being included as an axiom in ZF (or ZFC) and in some systems can even be derived.

    Does this notion ever become problematic outside of set theoretic terms (like for example in topology or field theory)?
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  6. #6
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    Quote Originally Posted by bleys View Post
    So this is all a matter of convention and convenience?


    Can I ask in what areas of math this is the case? I thought the empty set is fundamental from set theory, being included as an axiom in ZF (or ZFC) and in some systems can even be derived.

    Does this notion ever become problematic outside of set theoretic terms (like for example in topology or field theory)?

    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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  7. #7
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    Hi

    if , then and thus there's one unique function from or to the empty set.
    Perhaps you should say there is at most one function from A to B: for \emptyset to be a function from A to B, A has to be empty. We have:

    \text{if}\ A\neq\emptyset,\ \emptyset^A=\emptyset
    &
    \text{for all}\ B,\ B^\emptyset=1 . ( where 1 stands for \{\emptyset\} )
    Last edited by clic-clac; May 30th 2010 at 08:36 AM. Reason: definition
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    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by undefined View Post
    Here's some info on 0^0 from Wikipedia.
    Very interesting the 'history of different points of view' regarding 0^{0} described by Wiki!... in my opinion the Italian methematician Guglielmo Libri was right regarding the identity 0^{0}=1, even if he wasn't able to do a 'covincing demonstration' of that...

    Some year ago I found [at least that's my opinion...] a 'rigorous prove' that, if we indicate with \varphi(z) = z^{z} , is \varphi (0) = 1, exactly as for the Riemann zeta function \zeta(s) is \zeta (0) = -\frac{1}{2}. I am glad to supply here this 'demonstration', but with some preconditions ...

    Kind regards

    \chi \sigma
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  9. #9
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    Quote Originally Posted by clic-clac View Post
    Hi

    Perhaps you should say there is at most one function from A to B: for \emptyset to be a function from A to B, A has to be empty. We have:

    \text{if}\ A\neq\emptyset,\ \emptyset^A=\emptyset
    &
    \text{for all}\ B,\ B^\emptyset=1 . ( where 1 stands for \{\emptyset\} )

    I think in both cases we have the sets equaity A^\emptyset=\emptyset^B=\emptyset and thus in both cases we have one unique function, represented by the empty set as the unique subset of the corresponding cartesian product.

    Tonio
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  10. #10
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    Quote Originally Posted by Plato View Post
    I understand that is not practical in contemporary mathematics.
    R L Moore may have started the field of Point Set Topology. He certainly named it.
    His book FOUNDATIONS OF POINT SET THEORY contains not a single reference to an empty set.
    A point set cannot be empty as I said in my post.
    After rereading this thread, I can see wisdom in that position.
    Donít you think that these are ridiculous arguments?
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  11. #11
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    Quote Originally Posted by Plato View Post
    His book FOUNDATIONS OF POINT SET THEORY contains not a single reference to an empty set.
    That's strange. It doesn't even contain the word 'empty' (checked it on Google books just out of curiosity).
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  12. #12
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    Quote Originally Posted by TheCoffeeMachine View Post
    That's strange. It doesn't even contain the word 'empty' (checked it on Google books just out of curiosity).
    Well of course, point sets contain points.
    An empty set does not.
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  13. #13
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    Quote Originally Posted by Plato View Post
    Well of course, point sets contain points.
    An empty set does not.

    This is the point (pun intended): how does one define a "set point"? If one defines it as a set (like in standard set theory) containing at least one point then obviously the empty set is left out.
    Now, how can anyone do topology without the empty set is beyond my imagination, since from the very beginning one would have to leave out lots of things from "standard" topology.

    Tonio
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  14. #14
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    Quote Originally Posted by tonio View Post
    I think in both cases we have the sets equaity A^\emptyset=\emptyset^B=\emptyset and thus in both cases we have one unique function, represented by the empty set as the unique subset of the corresponding cartesian product.
    Tonio
    If you want to say the only function from A to B is the empty set, you must write B^A=\{\emptyset\}.

    Now, what about the functions from a non empty set A into B=\emptyset? Well by definition, if f\subseteq A\times B is a function, for all element a\in A, there must be an element b\in B such that (a,b)\in f. Since A\neq\emptyset, there is such a, but you cannot find any b\in B such that (a,b)\in f as B=\emptyset. Therefore there is no function from A to \emptyset, that is, \emptyset^A=\emptyset ( not \{\emptyset\} ).
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  15. #15
    MHF Contributor Swlabr's Avatar
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    The most reasonable way to study sets is as a category. If we view sets as a category, then there exists one function f: \emptyset \rightarrow \emptyset, the empty function, otherwise we cannot think of sets as a category.

    In fact, according to ZF we can form the set of function from the empty set to the empty set. This is most easily seen by looking at Lawvere's axioms which are slightly weaker than ZF. Two of these are,

    -There exists a set with no elements,
    -Given sets A and B we can form the set of functions from A to B.
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