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Breaking down a proof?

  1. Apr 10, 2012 #1
    This is the conclusion of a proof (any subset of a finite set is finite, which was worked on in this thread: https://www.physicsforums.com/showthread.php?t=592636) I recently presented, but I was told that I couldn't immediately make it:

    [itex]S = T \cup (S \setminus T)[/itex] (where T is a subset of S)
    [itex]f(S) = g(T) \cup h(S \setminus T)[/itex]

    I showed earlier in the proof that [itex]f(S) = S[/itex] and [itex]h(S \setminus T) = S \setminus T[/itex].
    I say that it immediately follows that [itex]g(T) = T[/itex] because

    [itex]f(x) = \left\{
    \begin{array}{l}
    g(x) & : x \in T\\
    h(x) & : x \in S \setminus T
    \end{array}
    \right.[/itex]

    which shows that [itex]g(T) \cap h(S \setminus T) = \emptyset[/itex].

    What could I be missing? Thanks for any help.
     
  2. jcsd
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