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Math Help - topology - the lift of a function f

  1. #1
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    topology - the lift of a function f

    Let X,Y be subsets of the universal set S. Suppose f:X to Y is a function. Define the lift of f to 2^X , F:2^X to 2^Y by F(A)=f(A), A in 2^X. Show the following:

    a) F is one-to-one if and only if f is one-to-one.
    b)F is onto if and only if f is onto.


    I know that 2^X is the power set of X, meaning that 2^X is the set of all subsets of X. Similarly, 2^Y is the set of all subsets of X. But I have no experience with lifts at all. To prove a), I will first assume that F is one-to-one. Then for a1,a2 in 2^X F(a1)=F(a2) implies a1=a2. But I don't understand how the lift is relating to f, so I cannot complete the proof. Thank you for your help.
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  2. #2
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    Quote Originally Posted by poXGxxi View Post
    Let X,Y be subsets of the universal set S. Suppose f:X to Y is a function. Define the lift of f to 2^X , F:2^X to 2^Y by F(A)=f(A), A in 2^X. Show the following:

    a) F is one-to-one if and only if f is one-to-one.
    Let A\subseteq X define f[A]=\{ y\in Y | y=f(x) \text{ for some }x\in A \}. I do not like to use 2^X because it does not usually mean the power set, instead I shall use \mathcal{P}(X) for the power set. Now define F: \mathcal{P}(X) \to \mathcal{P}(Y) by F(A) = f[A]. Assume that f is one-to-one and F(A) = F(B) then f[A] = f[ B ]. To show F is one-to-one we need to show A=B. Thus, let a\in A then f(a)\in f[A]. Since f[A] = f[ B ] it means f(a) \in f[ B ] and so by definition f(a) = f( b ) for some b\in B. However, f is one-to-one so a=b\in B. Thus, A\subseteq B, repeating the mirror argument shows B\subseteq A, which means A=B. Let us summarize what we did: we showed that if f is one-to-one then F is one-to-one.

    Conversely, say F is one-to-one, i.e. if F(A)=F(B) then A=B. We want to show f is one-to-one. Say that f(a) = f(b). Now form the sets \{ a \} and \{ b \} which are subsets of A. Note F( \{a\}) = \{ f(a)\} and F(\{b\}) = \{ f(b) \}. Therefore F( \{a \}) = F(\{ b\}) but by above assumptions it means \{ a \} = \{ b \} and so a=b. Let us summarize what we did: we showed that if F is one-to-one then f is one-to-one.

    Putting these together it means f is one-to-one if and only if F is one-to-one.

    You try the second part.
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