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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).

  2. jcsd
  3. Apr 1, 2013 #2


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


    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)##.

  4. Apr 1, 2013 #3
    The part you clarified was the part I was really struggling with. Thanks a lot!
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