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Showing that weak induction and strong induction are equivalent

  1. Dec 31, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that the principle of mathematical induction and strong induction are equivalent; that is, each can be shown to be valid from the other.

    2. Relevant equations

    Weak induction:
    [tex] (P(1) \wedge \forall k (P(k) \rightarrow P(k+1))) \rightarrow \forall n(P(n)) [/tex]
    Strong induction:
    [tex] (P(1) \wedge P(2) \wedge ... \wedge P(k) \wedge (P(k) \rightarrow P(k+1))) \rightarrow \forall n(P(n)) [/tex]

    3. The attempt at a solution

    First, I will try to show that, if weak induction is an axiom, then strong induction is always true. So, the following is always true:
    [tex] (P(1) \wedge \forall k (P(k) \rightarrow P(k+1))) \rightarrow \forall n(P(n)) [/tex]
    That is, if P(1) is true and P(k) implies P(k+1), then P(n) is true for all positive integers.
    For this, assume that P(1) is true and, for some k, P(k) → P(k+1) is true, then P(n) is true for all positive integers. So, for this value of k, P(1), P(2), ..., P(k) are all true. Then, the assertion [itex]P(1) \wedge P(2) \wedge...\wedge P(k) \rightarrow P(k+1)[/itex] is true. So,
    [tex] (P(1) \wedge P(2) \wedge ... \wedge P(k) \wedge (P(k) \rightarrow P(k+1))) \rightarrow \forall n(P(n)) [/tex]
    is always true.
    Is this part right?
     
    Last edited: Dec 31, 2011
  2. jcsd
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