1 / 14

CMSC 203 / 0201 Fall 2002

CMSC 203 / 0201 Fall 2002. Week #6 – 30 September / 2/4 October 2002 Prof. Marie desJardins. TOPICS. Proof methods Mathematical induction. MON 9/30 MIDTERM #1. Chapters 1-2. WED 10/2 PROOF METHODS (3.1). CONCEPTS / VOCABULARY. Theorems Axioms / postulates / premises

phuong
Télécharger la présentation

CMSC 203 / 0201 Fall 2002

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related