CMSC 203 / 0201 Fall 2002

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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 gt 1, then n2 gt
  1. Suppose that n2 gt 1. Then n gt 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 gt 3, then n2 gt
  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 gt 2, then n2 gt
  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(n1) is true
  for every integer n gt 1 (or n gt 0).
• Strong mathematical induction (second principle
  of mathematical induction)
• Inductive step Show that P(1) ? ? P(n) ?
  P(n1) 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.