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Math Help - a little bit of functions on sets

  1. #1
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    a little bit of functions on sets

    1. (a) For an arbitrary set Y , determine Y^\emptyset
    (b) For an arbitrary set X, determine  \emptyset ^X

    (a) I couldn't figure it out
    (b) I think it's the empty set but I can't show how
    if anyone could show me how I would greatly appreciate it
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by jbpellerin View Post
    1. (a) For an arbitrary set Y , determine Y^\emptyset
    (b) For an arbitrary set X, determine  \emptyset ^X

    (a) I couldn't figure it out
    (b) I think it's the empty set but I can't show how
    if anyone could show me how I would greatly appreciate it
    clarify what you mean. by the notation B^A, do you mean the set of functions from the set A to the set B?
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  3. #3
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    We can find disagreement on this question.
    Paul Halmos claims the following.
    (i) Y^\emptyset  has exactly one element, namely \emptyset, whether Y is empty or not and (ii) if X \ne \emptyset , then \emptyset ^X  = \emptyset".
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Plato View Post
    We can find disagreement on this question.
    Paul Halmos claims the following.
    (i) Y^\emptyset  has exactly one element, namely \emptyset, whether Y is empty or not and (ii) if X \ne \emptyset , then \emptyset ^X  = \emptyset".
    is my interpretation of the problem correct? in other words, is Halmos saying the set of function from the empty set to another set is empty, and thus, represented by the empty set?

    it seems strange that we should require X is nonempty in part (ii), since that is just a special case of part (i) (where Y is empty).

    how exactly is this claim justified? what are the other interpretations?
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  5. #5
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    Quote Originally Posted by Jhevon View Post
    how exactly is this claim justified? what are the other interpretations?
    All I can say is read his book: Naive Set Theory.
    I do not defend his answer.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Plato View Post
    All I can say is read his book: Naive Set Theory.
    I do not defend his answer.
    what are your views?

    is this one of those issues over which mathematicians are divided? one bunch accepts such a claim, and another bunch doesn't?
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  7. #7
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    Well I must say that my experience is different from the vast majority on mathematicians. In all my graduate training, my professors did not believe that a set could be empty. How can a point set be empty? If it is a point set then by definition it is not empty. Of course, that is a very narrow and not practical position.
    For me a function is a set of ordered pairs: f:A \mapsto B \Rightarrow \quad f \subseteq A \times B.
    I am not sure what \emptyset  \times Y\quad  \vee \quad Y \times \emptyset mean.

    In many ways, this is a pointless question.
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