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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

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CMSC 203 / 0201 Fall 2002

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Cmsc 203 0201 fall 2002 l.jpg

CMSC 203 / 0201Fall 2002

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

Prof. Marie desJardins


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TOPICS

  • Proof methods

  • Mathematical induction


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MON 9/30MIDTERM #1

Chapters 1-2


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WED 10/2PROOF METHODS (3.1)


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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


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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)


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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.”


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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.


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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.


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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.


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FRI 10/4MATHEMATICAL INDUCTION (3.2)


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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.


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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.


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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.


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