1. 2009/91 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. 2009/92 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. 2009/93 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. 2009/94 Formal Proofs A formal proof of a conclusion C, given premises p 1, p 2,…,p n consists 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. 2009/95 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. 2009/96 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. 2009/97 Proof Example cont. StepProved by 1.  sunny  cold Premise #1. 2.  sunny(p  q)  p “Simplification ” 3. swim  sunnyPremise #2. 4.  swim [  q  (p  q)]  p “Modus tollens” 5.  swim  canoe Premise #3. 6. canoe [p  (p  q)]  q “Modus ponens” 7. canoe  earlyPremise #4. 8. early [p  (p  q)]  q “Modus ponens”

8. 2009/98 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 n 2 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. 2009/99 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 0 a = 1 = 0 b.” 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. 2009/910 Proof by Contradiction 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.” 歸謬證法

11. 2009/911 Proving Existentials A proof of a statement of the form  x P(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. 2009/912 Limits on Proofs  Conjecture 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) 臆測；假說

13. 2009/913 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  (2 107,001 )  1. One of the most famous conjectures: (Fermat’s last Theorem) The equation x n + y n = z n has no nonzero integer solution whenever integer n  2. This theorem was proved by Wiles in 90s.