Methods of Proof (§1.7)

1. Methods of Proof (§1.7) • Methods of mathematical argument (proof methods) can be formalized in terms of rules of logical inference. • Mathematical proofs can themselves be represented formally as discrete structures. • We will review both correct & fallacious inference rules, & several proof methods. 推論規則

2. Proof Terminology • Theorem - A statement that has been proven to be true. • Axioms, postulates, hypotheses, premises - Assumptions (often unproven) defining the structures about which we are reasoning. • Rules of inference - Patterns of logically valid deductions from hypotheses to conclusions.

3. More Proof Terminology • Lemma - A minor theorem used as a stepping-stone to proving a major theorem. • Corollary - A minor theorem proved as an easy consequence of a major theorem. • Conjecture - A statement whose truth value has not been proven. (A conjecture may be widely believed to be true, regardless.)

4. Formal Proofs • A formal proof of a conclusion C, given premises p1, p2,…,pnconsists of a sequence of steps, each of which applies some inference to premises or previously-proven statements (as antecedents) to yield a new true statement (the consequent). • A proof demonstrates that if the premises are true, then the conclusion is true.

5. Formal Proof Example • Premises:“It is not sunny and it is cold.”“We will swim only if it is sunny.”“If we do not swim, then we will canoe.”“If we canoe, then we will be home early.” • Given these premises, prove “We will be home early” using inference rules.

6. Proof Example cont. • Let sunny=“It is sunny”; cold=“It is cold;” swim=“We will swim;” canoe=“We will canoe;” early=“We will be home early.” • Premises:(1) sunny  cold (2) swimsunny(3) swimcanoe (4) canoeearly 論證假設

7. Proof Example cont. StepProved by1. sunny  cold Premise #1.2. sunny (pq)p “Simplification”3. swimsunny Premise #2.4. swim[q(pq)]p “Modus tollens”5. swimcanoe Premise #3.6. canoe[p(pq)]q “Modus ponens”7. canoeearly Premise #4.8. early[p(pq)]q “Modus ponens”

8. Proof Methods For proving implications pq, we have: • Direct proof: Assume p is true, and prove q. Ex: “If n is an odd integer, then n2 is odd.” Pf: Assume that n = 2k + 1 for some integer k. Then …. • Indirect proof: Assume q, and prove p. Ex: “If 3n+2 is odd, then n is an odd integer.” Pf: Suppose that n is an even integer, n = 2k for some integer k. Then …. （直接證明） （間接證明；反證法）

9. Proof Methods (cont.) • Vacuous proof: “F  q” is always true. Ex: “If 0 > 1, then n > n+1.” • Trivial proof: “pT” is always true. Ex: “If a and b are positive integers, then 0a = 1 = 0b.” • Proof by cases: Show p(a  b) and (aq) and (bq). Ex: “If n is an integer, then n(n+1) is even.” （空泛證明） （平庸證明） （分案證明）

10. Proof by Contradiction 歸謬證法 • A method for proving p. • Assume p, and prove both q and q for some proposition q. • Thus p (q  q) • (q  q) is a trivial contradiction, equal to F • Thus pF, which is only true if p=F • Thus p is true. Ex: “There are infinite many primes.”

11. Proving Existentials • A proof of a statement of the form xP(x) is called an existence proof. • If the proof demonstrates how to actually find or construct a specific element a such that P(a) is true, then it is a constructive proof. • Ex: “If gcd(a, m)=1, then the inverse of a modulo m exists.” Pf: s, tZ such that sa+tm=1. Then s is the inverse of a modulo m. • Otherwise, it is nonconstructive. （存在證明） （建構性的） （非建構性的）

12. Limits on Proofs  Conjecture 臆測；假說 • Some very simple statements of number theory haven’t been proved or disproved! • E.g. Goldbach’s conjecture (1742): Every even integer n, n>2, is the sum of two primes. • n>2  primes p,q : n=(p+q)/2. (As of mid-2002, the conjecture has been checked for all positive even integer up to 4 following 14 zeros.) • “Every even positive integer n, n>2, is the sum of at most six primes.” (Ramaré, 1995)

13. More Conjectures • The Twin Primes Conjecture: “There are infinitely many twin primes.” - Twin primes: 3 and 5; 5 and 7; 11 and 13 etc. - The would record for twin primes, mid-2002, consists of the numbers 318,032,361(2107,001)1. • One of the most famous conjectures: (Fermat’s last Theorem) The equation xn+ yn = znhas no nonzero integer solution whenever integer n 2. • This theorem was proved by Wiles in 90s.