Methods of Proof

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Methods of Proof

• Introductory Discrete Mathematics (CS/MAT165)

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Try this

• Which of the following is true?
• For any real number x
• ?x - 1? ?x? - 1
• ?x - y? ?x? - ?y?
• Write it up and swap results with a neighbor.
• You have 5 minutes!

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Mathematical Proof

• Carefully reasoned argument
• Based on knowns

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Definitions

• Even
• n is even ? ?k, k ? Z such that n 2k
• Odd
• n is odd ? ?k, k ? Z such that n 2k 1
• Prime? Composite?

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Try this

• Prove that 2 is the only even prime number.
• Work in pairs
• Write it up
• Be prepared to defend your proof
• You have 15 minutes!

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Counterexample

• To prove p is True prove p is False
• To prove p is False prove p is True

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Try this

• ?a ? R, ? b ? R, (a2 b2)? (a b)
• Prove or disprove
• Work in pairs
• You have 10 minutes!

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Exhaustion

• ?n ? Z, if n is even and 4 ? n ? 30, then n can
  be written as a sum of two prime numbers
• Okay, lets try all the numbers!

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Direct Proof

• Express the statement as ?x ? D, if P(x) then
  Q(x)
• Pick an arbitrary x for which P(x) is true
• Show that Q(x) is true

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Try this

• Prove The sum of any two even integers is also
  an even integer

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Try this

• Prove The sum of any two even integers is also
  an even integer
• Proof
• Given even integers a and b
• By definition we can writea 2r, b 2s
• a b 2r 2s 2(r s) 2k
• ? (a b) is an even integer

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Writing Proofs

• Copy the statement of the theorem to be proved on
  your paper
• Clearly mark the beginning of your proof with the
  word Proof
• Make your proof self-contained
• Write your proof in complete sentences
• Give a reason for each assertion you make in your
  proof
• Include the little words that make the logic of
  your arguments clear

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Common Mistakes

• Arguing from examples
• Using the same identifier to mean two different
  things
• Jumping to a conclusion
• Begging the question assume what is to be proved
• Misuse of the word if

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Definitions

• Rational number
• A real number r is rational iff it can be
  expressed as a quotient of two integers with a
  nonzero denominator.
• r is rational ? ? integers a and b such that r
  a/b and b ? 0

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Try this

• Prove
• Every integer is a rational number
• The sum of two rational numbers is a rational
  number
• If r is rational then 2r is also rational
• The product of two rational numbers is a rational
  number

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Work time!

• p.155, Nos. 17-28
• p.170, Nos. 14-22
• p.178, Nos. 9-11
• p.179, Nos. 17-18
• Finish for homework