MATHEMATICAL PROOF

MATHEMATICAL PROOF

1. Contrapositive

   • P-->Q
     • assume Q not true
       • so P is not true
   • IF/THEN AND WHEN/MUST
   • EXAMPLE
     • If a product of two positive real numbers is greater than 100, then at least one of the number is greater than 10

2. Contradiction

   • broader use
     • EXAMPLE
       • For all integers n, if n2 is odd, then n is odd.
   • equally sensible proof
   • P-->Q
     • Q not true P true
       • but P is not true (negation)

3. Direct Proof

   • EXAMPLE
     • If n is any even integer, then (-1)^n = 1
   • most familiar forms of proof
   • use to prove P-->Q
     • assume P is true

4. Mathematical Induction

   • EXAMPLE
     • Prove that the sum of the first n natural numbers is given by this formula: 1+2+3+. . .+n = (n(n + 1))/2
   • Use when you have a proposition in a number sequence form
   • Domino Effect
     • When any domino falls, the next domino falls, so all domino will fall!

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