2. 資訊科學數學 7 : Proof, Number and Orders 陳光琦助理教授 (Kuang-Chi Chen) chichen6@mail.tcu.edu.tw

3. Basic Proof Methods

4. Nature & Importance of Proofs In mathematics, a proof is : In mathematics, a proof is : - a correct (well-reasoned, logically valid) and complete (clear, detailed) argument that rigorously & undeniably establishes the truth of a mathematical statement. - a correct (well-reasoned, logically valid) and complete (clear, detailed) argument that rigorously & undeniably establishes the truth of a mathematical statement. Why must the argument be correct & complete ? Why must the argument be correct & complete ? - Correctness prevents us from fooling ourselves. - Correctness prevents us from fooling ourselves. - Completeness allows anyone to verify the result. - Completeness allows anyone to verify the result.

5. Overview of Proofs Methods of mathematical argument (i.e., proof methods) can be formalized in terms of rules of logical inference. Methods of mathematical argument (i.e., proof methods) can be formalized in terms of rules of logical inference. Mathematical proofs can themselves be represented formally as discrete structures. Mathematical proofs can themselves be represented formally as discrete structures. We will focus on both correct & fallacious inference rules, & several proof methods. We will focus on both correct & fallacious inference rules, & several proof methods.

6. Applications of Proofs An exercise in clear communication of logical arguments in any area of study. An exercise in clear communication of logical arguments in any area of study. The fundamental activity of mathematics is the discovery and elucidation, through proofs, of interesting new theorems. The fundamental activity of mathematics is the discovery and elucidation, through proofs, of interesting new theorems. Theorem-proving has applications in program verification, computer security, automated reasoning systems, etc. Theorem-proving has applications in program verification, computer security, automated reasoning systems, etc. Proving a theorem allows us to rely upon on its correctness even in the most critical scenarios. Proving a theorem allows us to rely upon on its correctness even in the most critical scenarios.

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

8. Details of Assumption Axioms ( 公理 ) are statements that are taken as true that serve to define the very structures we are reasoning about. Axioms ( 公理 ) are statements that are taken as true that serve to define the very structures we are reasoning about. Postulates ( 假設 ) are assumptions stating that some characteristic of an applied mathematical model corresponds to some truth about the real world or some situation being modeled. Postulates ( 假設 ) are assumptions stating that some characteristic of an applied mathematical model corresponds to some truth about the real world or some situation being modeled.

9. Details of Assumption (cont ’ d) Hypotheses ( 假說 ) are statements that are temporarily adopted as being true, for purposes of proving the consequences they would have if they were true, leading to possible falsification of the hypothesis if a predicted consequence turns out to be false. Hypotheses ( 假說 ) are statements that are temporarily adopted as being true, for purposes of proving the consequences they would have if they were true, leading to possible falsification of the hypothesis if a predicted consequence turns out to be false. - Usually a hypothesis is offered as a way of explaining a known consequence. - Usually a hypothesis is offered as a way of explaining a known consequence. * Hypotheses are not necessarily believed or disbelieved except in the context of the scenario being explored. * Hypotheses are not necessarily believed or disbelieved except in the context of the scenario being explored.

10. Details of Assumption (cont ’ d) Premises ( 基本假設 ) are statements that the reasoner adopts as true so he can see what the consequences would be. Premises ( 基本假設 ) are statements that the reasoner adopts as true so he can see what the consequences would be. * Like hypotheses, they may or may not be believed. * Like hypotheses, they may or may not be believed.

11. More Proof Terminology Lemma ( 輔助定理 ) - A minor theorem used as a stepping-stone to proving a major theorem. Lemma ( 輔助定理 ) - A minor theorem used as a stepping-stone to proving a major theorem. Corollary ( 推論 ) - A minor theorem proved as an easy consequence of a major theorem. Corollary ( 推論 ) - A minor theorem proved as an easy consequence of a major theorem. Conjecture ( 推測 ) - A statement whose truth value has not been proven. (A conjecture may be widely believed to be true, regardless.) Conjecture ( 推測 ) - A statement whose truth value has not been proven. (A conjecture may be widely believed to be true, regardless.) Theory ( 理論 ) – The set of all theorems that can be proven from a given set of axioms. Theory ( 理論 ) – The set of all theorems that can be proven from a given set of axioms.

12. … more … Conjectures ( 推測 ) are statements that are proposed to be logical consequences of other statements, and that may be strongly believed to be such, but that have not yet been proven. Conjectures ( 推測 ) are statements that are proposed to be logical consequences of other statements, and that may be strongly believed to be such, but that have not yet been proven. Lemmas ( 輔助定理 ) are relatively uninteresting statements that are proved on the way (base) to proving an actual theorem of interest. Lemmas ( 輔助定理 ) are relatively uninteresting statements that are proved on the way (base) to proving an actual theorem of interest.

13. Graphical Visualization Various Theorems The Axioms of the Theory A Particular Theory A proof

14. Inference Rules - General Form Inference Rule Inference Rule - Pattern establishing that if we know that a set of antecedent statements of certain forms are all true, then a certain related consequent statement is true. - Pattern establishing that if we know that a set of antecedent statements of certain forms are all true, then a certain related consequent statement is true. antecedent 1 antecedent 2 …  consequent “  ” means “therefore” antecedent 1 antecedent 2 …  consequent “  ” means “therefore”

15. Inference Rules & Implications Each logical inference rule corresponds to an implication that is a tautology. Each logical inference rule corresponds to an implication that is a tautology. antecedent 1 Inference rule antecedent 2 …  consequent antecedent 1 Inference rule antecedent 2 …  consequent Corresponding tautology: Corresponding tautology: ((ante. 1)  (ante. 2)  …)  consequent

16. Some Inference Rules p Rule of Addition  p  q p Rule of Addition  p  q p  q Rule of Simplification  p p  q Rule of Simplification  p p Rule of Conjunction q  p  q p Rule of Conjunction q  p  q

17. Modus Ponens & Tollens pRule of modus ponens 斷言法 p  q (a.k.a. law of detachment)  q pRule of modus ponens 斷言法 p  q (a.k.a. law of detachment)  q  q p  q Rule of modus tollens 否定法  p  q p  q Rule of modus tollens 否定法  p “the mode of affirming” “the mode of denying” a.k.a. “also known as”

18. Syllogism Inference Rules p  qRule of hypothetical p  qRule of hypothetical q  r syllogism 三段論  p  r q  r syllogism 三段論  p  r p  qRule of disjunctive  p syllogism  q p  qRule of disjunctive  p syllogism  q Aristotle (ca. 384-322 B.C.)

19. 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 rule to premises or to previously-proven statements (as antecedents) to yield a new true statement (the consequent). 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 rule to premises or to 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. A proof demonstrates that if the premises are true, then the conclusion is true.

20. Formal Proof Example Suppose we have the following 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.” Suppose we have the following 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 the theorem “We will be home early” using inference rules. Given these premises, prove the theorem “We will be home early” using inference rules.

21. Proof Example (cont ’ d) Let us adopt the following abbreviations: Let us adopt the following abbreviations: sunny = “It is sunny”; cold = “It is cold”; swim = “We will swim”; canoe = “We will canoe”; early = “We will be home early”. sunny = “It is sunny”; cold = “It is cold”; swim = “We will swim”; canoe = “We will canoe”; early = “We will be home early”. Then, the premises can be written as: Then, the premises can be written as: (1)  sunny  cold (1)  sunny  cold (2) swim  sunny (2) swim  sunny (3)  swim  canoe (3)  swim  canoe (4) canoe  early (4) canoe  early

22. Proof Example (cont ’ d) StepProved by 1.  sunny  cold Premise #1. 2.  sunnySimplification of 1. 3. swim  sunnyPremise #2. 4.  swimModus tollens on 2,3. 5.  swim  canoe Premise #3. 6. canoeModus ponens on 4,5. 7. canoe  earlyPremise #4. 8. earlyModus ponens on 6,7.

23. Inference Rules for Quantifiers  x P(x)  P(o)(substitute any object o)  x P(x)  P(o)(substitute any object o) P(g)(for g a general element of u.d.)  x P(x) P(g)(for g a general element of u.d.)  x P(x)  x P(x)  P(c)(substitute a new constant c)  x P(x)  P(c)(substitute a new constant c) P(o) (substitute any extant object o)  x P(x) P(o) (substitute any extant object o)  x P(x)

24. Common Fallacies A fallacy is an inference rule or other proof method that is not logically valid. A fallacy is an inference rule or other proof method that is not logically valid. - May yield a false conclusion! - May yield a false conclusion! Fallacy of affirming the conclusion: Fallacy of affirming the conclusion: “p  q is true, and q is true, so p must be true.” ? “p  q is true, and q is true, so p must be true.” ? (No, because F  T is true.) (No, because F  T is true.) Fallacy of denying the hypothesis: Fallacy of denying the hypothesis: “p  q is true, and p is false, so q must be false.”? “p  q is true, and p is false, so q must be false.”? (No, again because F  T is true.) (No, again because F  T is true.)

25. Circular Reasoning The fallacy of (explicitly or implicitly) assuming the very statement you are trying to prove in the course of its proof. The fallacy of (explicitly or implicitly) assuming the very statement you are trying to prove in the course of its proof. Example: Prove that an integer n is even, if n 2 is even. Proof: “Assume n 2 is even. Then n 2 = 2k for some integer k. Dividing both sides by n gives n = (2k)/n = 2(k/n). So there is an integer j (namely k/n) such that n = 2j. Therefore n is even.” Begs the question: How do you show that j = k/n = n/2 is an integer, without first assuming n is even ???

26. A Correct Proof We know that n must be either odd or even. If n were odd, then n 2 would be odd, since an odd number times an odd number is always an odd number. Since n 2 is even, it is not odd, since no even number is also an odd number. Thus, by modus tollens, n is not odd either. Thus, by disjunctive syllogism, n must be even. ■ This proof is correct, but not quite complete, since we used several lemmas without proving them. Can you identify what they are?

27. A More Verbose Version Suppose n 2 is even  2|n 2  n 2 mod 2 = 0. Of course n mod 2 is either 0 or 1. If it’s 1, then n  1 (mod 2), so n 2  1 (mod 2), using the theorem that if a  b (mod m) and c  d (mod m) then ac  bd (mod m), with a=c=n and b=d=1. Now n 2  1 (mod 2) implies that n 2 mod 2 = 1. So by the hypothetical syllogism rule, (n mod 2 = 1) implies (n 2 mod 2 = 1). Since n 2 mod 2 = 0  1, by modus tollens we know that n mod 2  1. So by disjunctive syllogism we have that n mod 2 = 0  2|n  n is even. Uses some number theory we haven’t defined yet.

28. Divides, Factor, Multiple Let a,b  Z with a  0. Let a,b  Z with a  0. Def.: a|b  “a divides b” :  (  c  Z: b=ac) “There is an integer c such that c times a equals b.” Def.: a|b  “a divides b” :  (  c  Z: b=ac) “There is an integer c such that c times a equals b.” E.g., 3   12  True, but 3  7  False. E.g., 3   12  True, but 3  7  False. Iff a divides b, then we say a is a factor or a divisor of b, and b is a multiple of a. Iff a divides b, then we say a is a factor or a divisor of b, and b is a multiple of a. E.g., “b is even” :≡ 2|b. Is 0 even? Is −4? E.g., “b is even” :≡ 2|b. Is 0 even? Is −4?

29. Proof Methods for Implications For proving implications p  q, we have: Direct proof: Assume p is true, and prove q. Direct proof: Assume p is true, and prove q. Indirect proof: Assume  q, and prove  p. Indirect proof: Assume  q, and prove  p. Vacuous proof: Prove  p by itself. Vacuous proof: Prove  p by itself. Trivial proof: Prove q by itself. Trivial proof: Prove q by itself. Proof by cases: Show p  (a  b), and (a  q) and (b  q). Proof by cases: Show p  (a  b), and (a  q) and (b  q).

30. Direct Proof Example Definition: An integer n is called odd iff n = 2k+1 for some integer k; n is even iff n = 2k for some k. Axiom: Every integer is either odd or even. Theorem: (For all numbers n) If n is an odd integer, then n2 is an odd integer. Proof: If n is odd, then n = 2k+1 for some integer k. Thus, n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Therefore n2 is of the form 2j + 1 (with j the integer 2k2 + 2k), thus n2 is odd. □

31. Indirect Proof Example Theorem: (For all integers n) If 3n+2 is odd, then n is odd. Proof: Suppose that the conclusion is false, i.e., that n is even. Then n = 2k for some integer k. Then 3n+2 = 3(2k)+2 = 6k+2 = 2(3k+1). Thus 3n+2 is even, because it equals 2j for integer j = 3k+1. So 3n+2 is not odd. We have shown that ¬(n is odd)→¬(3n+2 is odd), thus its contrapositive (3n+2 is odd) → (n is odd) is also true. □ thus its contrapositive (3n+2 is odd) → (n is odd) is also true. □

32. Vacuous Proof Example Theorem: (For all n) If n is both odd and even, then n 2 = n + n. Proof: The statement “n is both odd and even” is necessarily false, since no number can be both odd and even. So, the theorem is vacuously true. □ So, the theorem is vacuously true. □

33. Trivial Proof Example Theorem: (For integers n) If n is the sum of two prime numbers, then either n is odd or n is even. Proof: Any integer n is either odd or even. So the conclusion of the implication is true regardless of the truth of the antecedent. Thus the implication is true trivially. □ Thus the implication is true trivially. □

34. Proof by Contradiction A method for proving p. A method for proving p. 1. Assume  p, and prove both q and  q for some proposition q. 2. Thus  p  (q   q). 3. (q   q) is a trivial contradiction, equal to F. 4. Thus  p  F, which is only true if  p=F. 5. Thus p is true.

35. Proof by Contradiction Example Theorem: 2 0.5 is irrational. Proof: Assume 2 1/2 were rational. This means there are integers i, j with no common divisors such that 2 1/2 = i/j. Squaring both sides, 2 = i 2 /j 2, so 2j 2 = i 2. So i 2 is even; thus i is even. Let i = 2k. So 2j 2 = (2k) 2 = 4k 2. Dividing both sides by 2, j 2 = 2k 2. Thus j 2 is even, so j is even. But then i and j have a common divisor, namely 2, so we have a contradiction. □

36. Review: Proof Methods So Far Direct, indirect, vacuous, and trivial proofs of statements of the form p  q. Direct, indirect, vacuous, and trivial proofs of statements of the form p  q. Proof by contradiction of any statement. Proof by contradiction of any statement. Next: Constructive and nonconstructive existence proofs. Next: Constructive and nonconstructive existence proofs.