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Math Help - problem involving composite functions

  1. #1
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    198

    problem involving composite functions

    Hi

    Here is a problem I am trying.

    Suppose A\neq \varnothing and

    f:\;A\longrightarrow A and

    \forall g[(g:A\longrightarrow A)\Rightarrow (f\circ g = f)]\cdots (1)

    I have to prove that f is a constant function.
    I will give the outline of what I did.
    Since \because A\neq \varnothing \Rightarrow \exists a \in A

    \therefore a \in A

    Now consider the set

    g=\{(x,a)\vert \; x\in A \; \}

    \because g\subseteq A\times A

    g is a relation from A to A. Then I proved that g is also a function by proving
    that

    \forall x \in A \exists ! b \in A ((x,b) \in g)

    next , I proved that g is a constant function by proving that

    \exists b \in A \forall x\in A (g(x)=b)

    and finally I used the given (1) for g , which implies that

    f\circ g =f

    To prove that f is a constant function, I have to prove

    \exists b\in A \forall x\in A (f(x)=b)

    since a\in A , I can associate some c\in A such that

    f(a)=c

    so Let b=c=f(a)

    so the goal now is

    \forall x\in A (f(x)=b)

    Let x be arbitrary , \therefore x\in A

    f(x)=f(g(x))=f(a)=b

    since x is arbitrary

    \therefore \exists b\in A \forall x\in A(f(x)=b)

    so f is a constant function..................

    is the proof too detailed ? since this is from Velleman's "how to prove it" , I think,
    author expects me to use all the logical machinery that I can use.

    correct ?
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  2. #2
    Senior Member
    Joined
    Feb 2010
    Posts
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    Thanks
    4

    Re: problem involving composite functions

    It's correct. Just as a matter of style though, it could be much more terse:

    Since A is non-empty, let a be in A.
    Let g = {<x a> | x is in A}.
    g is a function from A into A.
    Suppose x is in A.
    So f(x) = f(g(x)) = f(a).
    So f is a constant function.
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  3. #3
    Member
    Joined
    Oct 2010
    From
    Mumbai, India
    Posts
    198

    Re: problem involving composite functions

    thanks moeblee......
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