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CMSC 203 / 0201 Fall 2002

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CMSC 203 / 0201Fall 2002

Week #6 – 30 September / 2/4 October 2002

Prof. Marie desJardins

- Proof methods
- Mathematical induction

MON 9/30MIDTERM #1

Chapters 1-2

WED 10/2PROOF METHODS (3.1)

- Theorems
- Axioms / postulates / premises
- Hypothesis / conclusion
- Lemma, corollary, conjecture

- Rules of inference
- Modus ponens (law of detachment)
- Modus tollens
- Syllogism (hypothetical, disjunctive)
- Universal instantiation, universal generalization, existential instantiation (skolemization or Everybody Loves Raymond), existential generalization

- Fallacies
- Affirming the conclusion [abductive reasoning]
- Denying the hypothesis
- Begging the question (circular reasoning)

- Proof methods
- Direct proof
- Indirect proof, proof by contradiction
- Trivial proof
- Proof by cases
- Existence proofs (constructive, nonconstructive)

- Exercise 3.1.3: Construct an argument using rules of inference to show that the hypotheses “Randy works hard,” “If Randy works hard, then he is a dull boy,” and “If Randy is a dull boy, then he will not get the job” imply the conclusion “Randy will not get the job.”

- Exercise 3.1.11: Determine whether each of the following arguments is valid. If an argument is correct, what rule of inference is being used? If it is not, what fallacy occurs?
- (a) If n is a real number s.t. n > 1, then n2 > 1. Suppose that n2 > 1. Then n > 1.
- (b) The number log23 is irrational if it is not the ratio of two integers. Therefore, since log23 cannot be written in the form a/b where a and b are integers, it is irrational.
- (c) If n is a real number with n > 3, then n2 > 9. Suppose that n2 9. Then n 3.

- (Exercie 3.1.11 cont.)
- (d) A positive integer is either a perfect square or it has an even number of positive integer divisors. Suppose that n is a positive integer that has an odd number of positive integer divisors. Then n is a perfect square.
- (e) If n is a real number with n > 2, then n2 > 4. Suppose that n 2. Then n2 4.

- Exercise 3.1.17: Prove that if n is an integer and n3 + 5 is odd, then n is even using
- (a) an indirect proof.
- (b) a proof by contradiction.

FRI 10/4MATHEMATICAL INDUCTION (3.2)

- Proof by mathematical induction
- Inductive hypothesis
- Basis step: P(1) is true (or sometimes P(0) is true).
- Inductive step: Show that P(n) P(n+1) is true for every integer n > 1 (or n > 0).

- Strong mathematical induction (“second principle of mathematical induction”)
- Inductive step: Show that [P(1) … P(n)] P(n+1) is true for every positive integer n.

- Example 3.2.2 (p. 189): Use mathematical induction to prove that the sum of the first n odd positive integers is n2.
- Example 3.2.7 (p. 193): Use mathematical induction to show that the 2nth harmonic number,H2n = 1 + ½ + 1/3 + … + 1/(2n) 1 + n/2,whenever n is a nonnegative integer.

- Exercise 3.2.31:
- (a) Determine which amounts of postage can be formed using just 5-cent and 6-cent stamps.
- (b) Prove your answer to (a) using the principle of mathematical induction.
- (c) Prove your answer to (a) using the second principle of mathematical induction.