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Math Help - empty set and functions

  1. #1
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    empty set and functions

    Hi

    I was solving some problem about the set of functions and a doubt came to me (is it correct english ?). Consider function

    f: A\longrightarrow B

    Consider the case where A=\varnothing and B\neq \varnothing

    So A \times B = \varnothing

    Now function is type of a relation, and any subset of A \times B is a relation from A to B. Since A\times B =\varnothing , and

    \varnothing \subseteq \varnothing

     \varnothing is a relation from A to B. Now the definition of a function is

    \forall a  \in A \exists ! b\in B ((a,b) \in f)

    which can be written as an implication

    \forall a[(a\in A)\Rightarrow \exists ! b\in B ((a,b) \in f)]

    since A=\varnothing , the antecedent will be FALSE always , so the
    implication will be TRUE always, so the condition for the function is satisfied and
    we can say the the relation \varnothing is a function from A to B.
    \blacksquare

    is it correct reasoning ? \smile
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  2. #2
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    Re: empty set and functions

    is it correct reasoning ?
    You are absolutely right.

    To go a little further, consider the following definition. A set A is called initial if for every set B, there is one and only one function from A to B. Then the empty set is the unique initial set.
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  3. #3
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    Re: empty set and functions

    thanks makarov, I have finished first 4 chapters of Daniel Velleman's "How to prove it" and going to the fifth chapter , Functions. Since I am familiar with all the logical
    machinery , it was easy to draw the conclusion I drew. If somebody has only calculus background , then it will be difficult for that person to see why there is a function from A to B , when A is an empty set......... By the way , I just downloaded a beautiful book on set theory , "The Joy of sets:Fundamentals of contemporary set theory" by Keith Devlin and since I have already studied basic logic , its really joy to read it....

    \bigstar
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