InductionProofs

# InductionProofs - CMPS 101 Algorithms and Abstract Data...

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1 CMPS 101 Algorithms and Abstract Data Types Induction Proofs Let ) ( n P be a propositional function, i.e. P is a function whose domain is (some subset of) the set of integers and whose codomain is the set {True, False}. Informally, this means ) ( n P is a sentence, statement, or assertion whose truth or falsity depends on the integer n . Mathematical Induction is a proof technique which can be used to prove statements of the form ) ( : 0 n P n n (“for all n greater than or equal to 0 n , ) ( n P is true”), where 0 n is a fixed integer. A proof by Mathematical Induction contains two steps: I. Base Step: Prove directly that the proposition ) ( 0 n P is true. IIa. Induction Step: Prove )) 1 ( ) ( ( : 0 + n P n P n n . To do this pick an arbitrary 0 n n , and assume for this n that ) ( n P is true. Then show as a consequence that ) 1 ( + n P is true. The statement ) ( n P is often called the induction hypothesis , since it is what is assumed in the induction step. When I and II are complete we conclude that ) ( n P is true for all 0 n n . Induction is sometimes explained in terms of a domino analogy. Consider an infinite set of dominos which are lined up and ready to fall. Each domino is labeled by a positive integer, starting with 0 n . (It is often the case that 1 0 = n , which we assume here for the sake of definiteness). Let ) ( n P be the assertion: “the n th domino falls”. First prove ) 1 ( P , i.e. “the first domino falls”, then prove )) 1 ( ) ( ( : 1 + n P n P n which says “if any particular domino falls, then the next domino must also fall”. When this is done we may conclude ) ( : 1 n P n , “all dominos fall”. There are a number of variations on the induction step. The first is just a reparametrization of IIa. IIb. Induction Step: Prove )) ( ) 1 ( ( : 0 n P n P n n - > Let 0 n n > , assume ) 1 ( - n P is true, then prove ) ( n P is true. Forms IIa and IIb are said to be based on the first principle of mathematical induction . The validity of this principle is proved in the appendix of this handout. Another important variation is called the second principle of mathematical induction , or strong induction . IIc. Induction Step: Prove )) 1 ( )) ( : (( : 0 + n P k P n k n n Let 0 n n , assume for all k in the range n k n 0 that ) ( k P is true. Then prove as a consequence that ) 1 ( + n P is true. In this case the term induction hypothesis refers to the stronger assumption: ) ( : k P n k . The strong induction form is often reparametrized as in IIb :

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2 IId. Induction Step: Prove )) ( )) ( : (( : 0 n P k P n k n n < > Let 0 n n > , assume for all k in the range n k n < 0 , that ) ( k P is true, then prove as a consequence that ) ( n P is true. In this case the induction hypothesis is ) ( : k P n k < . In terms of the Domino analogy, the strong induction form IId says we must show: (I) the first domino
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InductionProofs - CMPS 101 Algorithms and Abstract Data...

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