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CSCB36H3 (2)
Lecture

Day 1 Sep 12

3 Pages
111 Views

Department
Computer Science
Course Code
CSCB36H3
Professor
Nick Cheng

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Description
CSCB36 Nick Cheng Chapter 1: Natural Numbers: ℕ = { 0, 1, 2, … } Predicates on Natural Numbers: P 1n): ∑n=0n = n(n+2) / 2 P (n): there are n students in B36 2 Let P be a predicate on ℕ. Principle of Simple Induction (PSI): Suppose P(0) holds, For all n ≥ 0, [P(n) => P(n+1)] Suppose also for any n ≥ 0, if P(n) holds, then P(n+1) holds. Then we can conclude: P(n) holds for all n ≥ 0. P(0), P(0) => P(1), P(1) => P(2), … Principle of Complete Induction (PCI): Suppose P(0) holds, Suppose for any n > 0, if P(j) holds whenever 0 ≤ j < n, then P(n) Then we can conclude: P(n) holds for all n ≥ 0. P(0), P(0) => P(1), P(0) and P(1) => P(2), P(0) + (P(1) + P(2) => P(3), … Principle of Well Ordering (PWO): Suppose A ⊆ ℕ and A ≠ ∅ Then we can conclude: A has a minimum element i.e a # m s.t m ≤ m’ for all m’ ∈ A Proofs: e.g 1.7: Stamps: P(n): n cents of postage can be made using only 4c & 7c stamps. n = K * 4 + L *
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