EECS
lec12-induction

# lec12-induction - EECS 203 Winter 2007 Discrete Mathematics...

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EECS 203, Winter 2007 Discrete Mathematics Lecture 12 Mathematical Induction February 13 Reading: Rosen [4.1, 4.2] February 13 Mathematical Induction, Page 1

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12.1 The Principle of mathematical induction To prove that P ( n ) is true for all n 0, Basis Step : Prove that P (0) is true; Inductive Step : Prove that if P ( k ) is true for some integer k 0, then P ( k + 1) is true. Axiom of number theory. If want to prove P ( n ) is true for all n n 0 , Basis Step . Prove that P ( n 0 ) is true, Inductive Step . Prove that if P ( k ) is true for some k n 0 , then P ( k + 1) is also true. February 13 Mathematical Induction, Page 2
12.2 Example: Inclusion-Exclusion Principle Theorem. Let A 1 , · · · , A n be finite sets. Prove that | A 1 A 2 ∪ · · · ∪ A n | = n i =1 ( - 1) i - 1 1 j 1 <j 2 < ··· <j i n | A j 1 A j 2 ∩ · · · ∩ A j i | . Proof. We prove by induction on n . Basis Step . n = 1, L.H.S. = | A | = R.H.S.. Thus P (1) is true. Inductive Step . Suppose P ( k ) is true for some k 1. Consider n = k + 1. Set A k = A k A k +1 . Apply P ( k ) on A 1 , ..., A k - 1 , A k ,

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