Results 1 to 5 of 5

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

  1. #1
    Junior Member
    Joined
    Sep 2010
    Posts
    32

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

    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))))

    My attempt:

    a) first, g^(-1)2(B) = {x belongs toA2 : g2(x) belongs to B}

    --> g^(-1)(g^(-1)2(B)) = g^(-1)1({x belongs to A2 : g2(x) belongs to B})

    --> g^(-1)1({x belongs to A2 : g2(x) belongs to B}) = {y belongs to A1 : g1(y) belongs to {x belongs to A2 : g2(x) belongs to B}}

    -->g^(-1)1(g^(-1)2(B)) = {y belongs to A1 : (g2 ◦ g1)^(-1)(y) belongs to 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
    Last edited by habsfan31; September 17th 2010 at 05:57 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Aug 2010
    Posts
    37
    I guess you mean \ {g_1}^{-1}( {g_2}^{-1} (B)) = \{ x \in A_1 : g_1(a) \in {g_2}^{-1} (B) \} and usually one proves that two sets are equal if each one is contained in the other.
    Yes induction is the best way to go about this.
    Let \ h = f_{n}\circ f_{n-1} .... \circ f_2 \circ f_1
    Now you know what you want to prove holds for n =2 and you are assuming by hypothesis that \ (f_{n} \circ ... \circ f_1)^{-1} = {f_1}^{-1}(....({f_n}^{-1} (B) )
    so \forall   A \subset A_n ;  h^{-1} (A)  = {f_1}^{-1}(....({f_n}^{-1} (A) )
    hope this helps
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2010
    Posts
    32
    Thanks for the reply. Im still not quite getting it. How would you start the induction?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Aug 2010
    Posts
    37
    for n=2, you proved that \ (f_{2} \circ  f_1)^{-1} (B) = {f_1}^{-1}({f_2}^{-1} (B) ) for \ B  \subset A_2
    Let n \ge 2 be such that  (f_{n} \circ ... \circ f_1)^{-1} = {f_1}^{-1}(....({f_n}^{-1} (B) ) holds true for \ B  \subset A_n.
    We want :  (f_{n+1} \circ ... \circ f_1)^{-1} = {f_1}^{-1}(....({f_{n+1}}^{-1} (B) ) for \ B  \subset  A_{n+1}
    Let \ h = {f_n}\circ .... \circ {f_1}
    \ h^{-1} (A) = {f_1}^{-1}(....({f_n}^{-1} (A) ) for \  A  \subset A_n by hypothesis
    \ f_{n+1} : A_n \rightarrow A_{n+1}
    \ f_{n+1} \circ ... \circ f_1 (a) = f_{n+1} \circ h (a) \forall   a  \in  A_1
    for \   B  \subset A_{n+1}  , (f_{n+1} \circ f_{n} \circ... \circ f_1)^{-1}) (B) =  (f_{n+1} \circ h)^{-1} (B)
    Now by proof for n = 2 , you know
    \ (f_{n+1} \circ  h)^{-1} (B) = {h}^{-1}({f_{n+1}}^{-1} (B) )
    and we know that \ h^{-1} (A) = {f_1}^{-1}(....({f_n}^{-1} (A) ) for \  A  \subset A_n
    combining the last two lines you get that  (f_{n+1} \circ ... \circ f_1)^{-1} (B) = {f_1}^{-1}(....({f_{n+1}}^{-1} (B) ) for \ B  \subset  A_{n+1}
    Last edited by bubble86; September 19th 2010 at 10:10 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Sep 2010
    Posts
    32
    Thanks so much, i get it now!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. why does lim(z→e^(πi/4)) (z - e^(πi/4))/(1 + z^4) =...
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: September 8th 2011, 05:51 AM
  2. Replies: 6
    Last Post: March 2nd 2011, 12:36 AM
  3. is f: x → |x| a function of x?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: June 3rd 2010, 09:09 AM
  4. A ⊂ b ⇒ p (a) ≤ p (b)
    Posted in the Advanced Statistics Forum
    Replies: 7
    Last Post: February 16th 2010, 01:18 PM
  5. proof of a limit as n→∞
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 21st 2009, 03:26 PM

Search Tags


/mathhelpforum @mathhelpforum