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Properties of inverse image?

  1. Sep 19, 2011 #1
    I am studying the very first chapter of analysis, but can't quite get through this problem:

    Prove f −1(f(A)) ⊇ A
    Prove f(f −1(B)) ⊆ B
     
  2. jcsd
  3. Sep 23, 2011 #2

    Stephen Tashi

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    How did you begin the problem?
    For [itex] A \subseteq f^{-1}( f(A)) [/itex], consider two cases.
    case 1:[itex] A = \emptyset [/itex]
    case 2: [itex] A \neq \emptyset [/itex]
     
  4. Sep 24, 2011 #3
    in general , to prove that [tex]A\subseteq B[/tex] , we prove

    [tex]\forall x[x\in A\Rightarrow x\in B][/tex]


    and to prove this ,you let x be arbitrary. since we have an implication inside the square bracket, we assume antecedent , and set out to prove consequent . So assume
    [tex]x\in A[/tex]
     
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