Induction

1
Induction
2
Proof by Induction

• To prove ?n?N P(n)
• Do the base case (or basis step)
• Prove P(0).
• Do the inductive step
  Prove ?k?N P(k)?P(k1).
• Let k?N, assume P(k). (inductive hypothesis)
• Now, under this assumption, prove P(k1).
• The inductive inference rule then gives us?n?N
  P(n).

3
Generalization

• Induction can also be used to prove ?n?c P(n)
  for a given constant c?Z, where c may not be 0.
• In this circumstance, the base case is to prove
  P(c) rather than P(0), and the inductive step is
  to prove ?n?c (P(n)?P(n1)).

4
Proof by Induction

• Show Sc nc P(c)
• Assume SK nk P(k)
• Show SK1 nk1 P(k1)
• ? ?n?c P(n)

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pg 117 --- Summations 3.3 ca cb
c(ab)(Distributive Law) 3.4
abcd ab cd (Associative Law) add
3.7 Sia to b Si1 to b - Si1 to
a-1 Often use Si1 to k1 Si1 to k Sik1
to k1
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Si1 to n ( i ) (i1)/2 1 2 3 4 5 6 7 8 9
10 i/2 pairs that sum to i1