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: Mathematical Reasoning and Writing:

  1. Apr 1, 2013 #1
    1. The problem statement, all variables and given/known data

    Let f : A → B be a function and let S, T [itex]\subseteq[/itex] A and U, V [itex]\subseteq[/itex] B:

    Prove that if U [itex]\subseteq[/itex] V; then f-1(U) [itex]\subseteq[/itex] f-1(V):

    2. Relevant equations

    Preimage: f-1(U) = {x [itex]\in[/itex] f-1(U) [itex]\ni[/itex] f(x) [itex]\subseteq[/itex] U}

    3. The attempt at a solution

    Assume U [itex]\subseteq[/itex] V.

    Let x [itex]\in[/itex] f-1(U), then f(x) = y [itex]\subseteq[/itex] U.

    Since U [itex]\subseteq[/itex] V and y [itex]\in[/itex] U, then y [itex]\in[/itex] V.

    Then by f-1(V), for y [itex]\in[/itex] V, x [itex]\subseteq[/itex] f-1(V).

    Therefore, f-1(U) [itex]\subseteq[/itex] f-1(V).

    Q.E.D.
     
  2. jcsd
  3. Apr 1, 2013 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Should be ##f(x) = y \in U##.

    Yes.

    Right idea, but stated poorly. It is not clear what the words "by" and "for" are attempting to convey here. And again you used ##\subseteq## when you should have used ##\in##.

    Here is a clearer and simpler statement: ##y = f(x) \in V##, so ##x \in f^{-1}(V)##.

    Correct.
     
  4. Apr 1, 2013 #3
    The part you clarified was the part I was really struggling with. Thanks a lot!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted