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Math Help - Composition as a Sets of Ordered Pairs

  1. #1
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    Composition as a Sets of Ordered Pairs

    I've come across a challenging problem in my test review:

    Given f = {(a, b), (b, a), (c, b)}, a function from X = {a, b, c} to X:

    (a) Write f o f and f o f o f as sets of ordered pairs.
    (b) Define f^n = f o f o ... o f to be the n-fold composition of f with itself. Write f^9 and f^623 as sets of ordered pairs.

    I have no idea how to approach this. I know that f o f and f o f o f are f(f) and f(f(f)), but I can't see how I can compose that. Where in f can I insert f? Part (b) also looks daunting.

    Any help would be appreciated!
    Last edited by john192; September 30th 2009 at 08:15 PM.
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  2. #2
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    I think I've figured out (a):

    f(a) = b; f(b) = a; f(c) = b, so

    f(f) = {(b, a), (a, b), (b, a)}, and
    f(f(f)) = {(a, b), (b, a), (c, a)}

    But I'd love for someone to confirm this.

    I'm still having trouble with part (b) also. Thanks for any help!
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  3. #3
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    You need ti rethink the whole problem.
    Here is how it works.
    f \circ f(a) = f(f(a)) = f(b) = a\, \Rightarrow \,(a,a) \in f \circ f

    \begin{gathered}<br />
  f \circ f = \left\{ {(a,a),(b,b),(c,a)} \right\} \hfill \\<br />
  f \circ f \circ f = \left\{ {(a,b),(b,a),(c,b)} \right\} \hfill \\<br />
  f \circ f \circ f \circ f = \left\{ {(a,a),(b,b),(c,a)} \right\} \hfill \\ <br />
\end{gathered}
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