Proof Methods: Part 3

1
Proof Methods Part 3

• Sections 3.1-3.6

2
Another Direct Proof

• Prove The sum of two rational numbers is a
  rational number
• Suppose Let s and t be rational numbers
• Show The sum of s and t is also rational
• s a/b, ?a,b ?Z defn of rational
• t c/d, ?c,d ?Z defn of rational
• st a/b c/d (adcb)/bd common denom
• Let p adcb and q bd be ints add mult of
  ints
• q ? 0 zero product property
• ? (adcb)/bd is rational by defn

3
Proof by Contradiction

• Prove The sum of any irrational number and any
  rational number is irrational
• Negating gives
• The sum of an irrational number and a rational
  number is rational
• Suppose Let q be an irrational number and r be a
  rational number
• Show The sum of q and r is rational
• r a/b, ?a,b ?Z and b ? 0 defn of rational
• rq c/d, ?c,d ?Z and d ? 0 defn of rational
• rq a/b q c/d substitution
• q c/d a/b subtraction
• q (bc ad)/bd common denom
• Hence q is the quotient of two integers (bc ad)
  and bd, and bd ? 0
• ? Thus by defn of rational, q is rational which
  contradicts the original supposition that it is
  irrational. Hence the supposition is false and
  the Theorem is True.

4
Writing Good Proofs

• Copy the Theorem to be proved
• Rewrite into ? and ? form using predicates and
  connectives
• Identify the Starting Point (what you assume to
  be True) and the Conclusion to be Shown (what
  you are trying to prove)
• Clearly state Suppose and Show
• Write your proof in complete sentences
• Use symbols and abbreviations as needed
• Give a reason for each assertion you make in your
  proof
• By hypothesis, by definition, by Theorem,
• Include the little words that make the logic of
  your arguments clear
• Let, since, thus, so, it follows that, ..

5
Common Mistakes

• Arguing from examples
• Look for examples, but you can not prove a
  general statement by showing it to be true for
  some special cases
• Using the same letter to mean two different
  things
• Suppose m and n are odd integers, thus m2k1 and
  n2k1 for some int k
• This implies that m2k1n (i.e., m and n are
  ints that equal each other)
• Jumping to conclusions
• Alleging the truth of something without giving an
  adequate reason
• Begging the question
• Assuming what is to be proved (instead of putting
  it in Show)
• Misuse of the word If
• Using If instead of because reflects
  imprecise thinking that sometimes leads to
  problems

6
Proof?

• Prove if n3 is even then n is even
• Proof Assume n3 is even.
• Then ?k?I n3 8k3 for some integer k. It
  follows that n 3?8k3 2k. Therefore n is
  even.
• Statement is true but argument is false.
• Argument assumes that n is even in making the
  claim n38k3, rather than n3 2k.