1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Preimage and function composition?

  1. Sep 18, 2012 #1
    1. The problem statement, all variables and given/known data

    Let f:S→T and let A[itex]\subseteq[/itex]T. Define the preimage of A as f-1(A)={x in S: f(x) is in A}.
    Demonstrate that for any such map f and B[itex]\subseteq[/itex]ran(f), f(f-1(B)) = B.

    I am going to use set inclusion to prove this, but can I use function composition in the portion in red? I was going to say an element y is in f(f-1(B)) and then was thinking to apply function composition so as to map an element x back to S then apply f to it and map the new element to B.
    Does this seem right?

  2. jcsd
  3. Sep 18, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It's not exactly function composition, because [itex]f^{-1}[/itex] isn't necessarily a function. This is because, for a given [itex]y[/itex] in the image of [itex]f[/itex], there may be more than one [itex]x[/itex] satisfying [itex]y = f(x)[/itex].

    The way to interpret the expression in red is that [itex]f^{-1}(B)[/itex] is the set of all [itex]x[/itex] such that [itex]f(x) \in B[/itex]. And f(f^{-1}(B)) is the image of that set under [itex]f[/itex].
  4. Sep 19, 2012 #3
    For what it's worth, it's good to remember that the notation f-1 is actually being abused here. This is the normal use, but it's good to remember what it means.

    f is a function from S -> T; and at some other time in class we may have occasion to talk about its inverse function f-1 : T -> S.

    However here we're not doing that. f-1 is being defined as a function from the power set of T to the power set of S; that is,

    f-1 : P(T) -> P(S)

    For each subset of T, we define P(T) as a particular subset of S.

    You have to be careful to remember that f goes left to right from points to points, and f-1 goes from right to left from subsets to subsets.
  5. Sep 19, 2012 #4
    I see, thank you! I was a little hesitant with using composition, but now I know to just use the definition of preimage and I think that should do the trick.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook