Proof Reprise

  1. Proof that k divides N: "show me the k!"
  2. Inductive proof: "Find me the Induction Hypothesis!"
  3. Prove that you can win a game: "Tell me what move I need to play!"

Induction Proof Reprise

  1. Base Case, Induction Hypothesis, Induction Proof, Conclusion
  2. Strong Induction
  3. Double Induction
Induction.
 The form of argument for induction is:
P(1)
forall k, P(k) --> P(k+1)
-------------------------
forall n>0, P(n).

So, for kinds of problems can induction work for? It can work on natural numbers, or integers greater than zero. Can it work on real numbers? Can it work on rational numbers?

You can induct over any set that is "Well ordered". A set is "well ordered" if any finite subset of the set has an minimum element.

  1. The normal ordering <= on the natural numbers is a well ordering.
  2. The normal ordering <= on the integers is not a well ordering.
  3. The normal ordering <= on the reals is not a well ordering. (Whether or not there exists a well ordering on the reals is complicated)
One way to think about this is to ask if there is any "infinite"...
Warm up

Ok, so now let's do a quick warm up problem ...a generic induction problem that might (some day) let you solve a problem quickly. This also introduces us to the summation notation: 

Prove: a0 + a1 + a2 + an =
      (an+1 - 1) / (a-1)


 strong induction

The form of argument for strong induction is:

P(1)
forall k, (P(1) ^ P(2) ^ P(3) ^ ... P(k-1) ^ P(k)) --> P(k+1)
-------------------------
forall n, P(n).



That is, when we prove P(k+1), we get to use any of the P(??) that
come before.