lec12-induction - EECS 203, Winter 2007 Discrete...

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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
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12.2 Example: Inclusion-Exclusion Principle Theorem. Let A 1 , ··· , A n be finite sets. Prove that | A 1 A 2 ∪ ··· ∪ A n | = n X i =1 ( - 1) i - 1 X 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
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This note was uploaded on 04/01/2008 for the course EECS 203 taught by Professor Yaoyunshi during the Winter '07 term at University of Michigan.

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lec12-induction - EECS 203, Winter 2007 Discrete...

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