CMSC 203 / 0201 Fall 2002

1. CMSC 203 / 0201Fall 2002 Week #6 – 30 September / 2/4 October 2002 Prof. Marie desJardins

2. TOPICS • Proof methods • Mathematical induction

3. MON 9/30MIDTERM #1 Chapters 1-2

4. WED 10/2PROOF METHODS (3.1)

5. CONCEPTS / VOCABULARY • 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

6. CONCEPTS / VOCABULARY II • 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)

7. Examples • 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.”

8. Examples II • 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.

9. Examples III • (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.

10. Examples IV • 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.

11. FRI 10/4MATHEMATICAL INDUCTION (3.2)

12. CONCEPTS/VOCABULARY • 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.

13. Examples • 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.

14. Examples II • 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.