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Induction proof notation

  1. Aug 27, 2006 #1
    Hi, I have to write several inductive proofs for a class.

    Typically, 'n' is used to denote the iterative variable in the problem statement. Then I show the case for n = 1 (or however appropriate for the proof) then proceed to show that if valid for n = k, then valid for n = k+1

    However, there are more variables now in the given problem statement (using variables n, k, m, etc) which leaves me uncertain as to how to properly label the inductive step variable. I feel re-using n, k, or m would create additional confusion, as it would if I used alternative variable labels that are not traditionally used to refer only to integers (a, b, c, x, y, z, etc)

    How should I alleviate this? Am I clear enough? It's kind of hard to describe...
     
  2. jcsd
  3. Aug 27, 2006 #2
    I don't think it will really make a difference as long as it is clear from your proof how you are using the variable, but I guess you could use j, or l if you want to stick the the letters of the alphabet surrounding n, k, and m.
     
  4. Aug 27, 2006 #3

    0rthodontist

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    You can also use n for the induction variable, or if not n, whatever variable you happen to be inducting on. It's unnecessary complication to use k in the first place. You just argue:
    Code (Text):

    Assume S(n)
       ...
       S(n+1)
    S(n) --> S(n+1) (conditional proof)
    for all n, S(n) --> S(n+1) (universal generalization)
     
    Since n is bound by a quantifier outside of the conditional proof, there is no scope conflict.
     
    Last edited: Aug 27, 2006
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