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Math Help - Proving the Hypothetical Syllogism using Induction

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    Proving the Hypothetical Syllogism using Induction

    Hi All,

    First post here. thanks for the help in advance!

    Okay first, the problem:

    In Logic, the hypothetical syllogism says that given three statements, P, Q, and R, the hypothesis P \rightarrow Q and Q \rightarrow R logically imply P \rightarrow R. Symbolically written ((P \rightarrow Q) ^ (Q \rightarrow R) \Rightarrow (P \rightarrow R))
    Use induction to show that, given n statements ( ^A{1}, ^A{2}, .... ^A{n}) that ( ^A{1} \rightarrow ^A{2}) ^ ( ^A{2} \rightarrow ^A{3} ^ ... ^ ( ^A{n-1} \rightarrow ^A{n}) \Rightarrow ( ^A{1} \rightarrow ^A{n}) for n \geq 2


    The problem I am having is that I am not sure how to work out an equation from this statement that I can start to use induction on. Also, I'm not sure if I should try to prove this using an inequality expression or an = expression. It would seem to me that an inequality would make more sense in an implication of this sort, but I am anything but positive on it.

    I tried subbing in an arbitrary x for ^A{1}, and assigning (x+1) to ^A{2}, and using ^A{1} \leq ^A{2}, but my problem exploded and became unreasonable.

    Can someone please help me find what recursive function or sum equation I should start with? I should be able to solve the proof from there with mathematical induction (also, do you suppose strong induction or standard will work in this case?)

    Thanks again all!

    -EDIT-

    Here is what I have done so far. I think it proves this is true for ^A{n} when f(n)=n, but is this all I need to do? What about cases where f(n) is equal to something other than just n?

    let ^A{n} = f(n) where:
    f(1) = 1
    f(n) = n for n \geq 2

    Step 1: (Base case) LHS ^A{2} = 2, and RHS f(2) = 2
    Step 2: (Show ^A{n+1} = f(n+1)
    ^A{n+1} = f(n+1)
    ^A{n+1} = n+1 \leftarrow induction step

    so there is the proof for ^A{n+1}, but something just feels wrong about this. have I solved this problem, or am I missing something here?
    Last edited by TravisH82; September 30th 2009 at 09:45 AM. Reason: Showing my work so far
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