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Homework Help: Analysis help->Suppose that g1 : A1 → A2 and g2 : A2 → A3. If B ⊂ A3, show that:

  1. Sep 17, 2010 #1
    Analysis help-->Suppose that g1 : A1 → A2 and g2 : A2 → A3. If B ⊂ A3, show that:

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

    Suppose that g1 : A1 → A2 and g2 : A2 → A3. If B ⊂ A3, show that
    a) (g2 ◦ g1)^(−1)(B) = g^(−1)1 (g^(−1)2 (B)).

    b) Now suppose that n ≥ 2 and gi : Ai → Ai+1 for i = 1, 2, . . . n. If B ⊂ An+1 show that
    (gn ◦ gn−1 ◦ . . . ◦ g2 ◦ g1)^(−1)(B) = g^(−1)1 (g^(−1)2 (. . . (g^(−1)n (B))))

    2. The attempt at a solution

    a) first, g^(-1)2(B) = {x[tex]\in[/tex]A2 : g2(x)[tex]\in[/tex]B}

    --> g^(-1)(g^(-1)2(B)) = g^(-1)1({x [tex]\in[/tex] A2 : g2(x) [tex]\in[/tex] B})

    --> g^(-1)1({x [tex]\in[/tex] A2 : g2(x) [tex]\in[/tex] B}) = {y [tex]\in[/tex] A1 : g1(y) [tex]\in[/tex] {x [tex]\in[/tex] A2 : g2(x) [tex]\in[/tex] B}}

    -->g^(-1)1(g^(-1)2(B)) = {y [tex]\in[/tex] A1 : (g2 ◦ g1)^(-1)(y) [tex]\in[/tex] B}

    Which by definition = ((g2 ◦ g1)^(−1)(B)

    b) im not sure how to do part B, can anybody help me. Would i need to use induction? I apologize if this all looks confusing, i tried to make it as clear as possible.

    Thanks
     
  2. jcsd
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