資訊科學數學資訊科學數學 7 7 ::

Proof, Number and OrdersProof, Number and Orders

陳光琦助理教授 陳光琦助理教授 (Kuang-Chi Che(Kuang-Chi Chen)n)

chichen6@mail.tcu.edu.twchichen6@mail.tcu.edu.tw

Basic Proof MethodsBasic Proof Methods

Nature & Importance of Nature & Importance of ProofsProofs

• In mathematics, a In mathematics, a proofproof is : is :

- a - a correctcorrect (well-reasoned, logically valid) and (well-reasoned, logically valid) and completecomplete (clear, detailed) argument that (clear, detailed) argument that rigorously & undeniably establishes the truth of rigorously & undeniably establishes the truth of a mathematical statement.a mathematical statement.

• Why must the argument be Why must the argument be correct & complete correct & complete ??

- - CorrectnessCorrectness prevents us from fooling ourselves. prevents us from fooling ourselves.

- - CompletenessCompleteness allows anyone to verify the result. allows anyone to verify the result.

Overview of ProofsOverview of Proofs

• Methods of mathematical argument (Methods of mathematical argument (i.e.i.e., proof , proof methods) can be formalized in terms of methods) can be formalized in terms of rules rules of logical inferenceof logical inference..

• Mathematical Mathematical proofsproofs can themselves be can themselves be represented formally as represented formally as discrete structuresdiscrete structures..

• We will focus on both We will focus on both correctcorrect & & fallaciousfallacious inference rules, & several proof methods.inference rules, & several proof methods.

Applications of ProofsApplications of Proofs

• An exercise in clear communication of An exercise in clear communication of logicallogical arguments in any area of study.arguments in any area of study.

• The fundamental activity of mathematics is the The fundamental activity of mathematics is the discoverydiscovery and and elucidationelucidation, through proofs, of , through proofs, of interesting new theorems.interesting new theorems.

• Theorem-provingTheorem-proving has applications in program has applications in program verification, computer security, automated verification, computer security, automated reasoning systems, reasoning systems, etc.etc.

• Proving a theorem allows us to rely upon on its Proving a theorem allows us to rely upon on its correctnesscorrectness even in the most critical scenarios. even in the most critical scenarios.

Proof TerminologyProof Terminology

• Theorem (Theorem ( 定理定理 ))

- A statement that has been - A statement that has been provenproven to be true. to be true.

• Axioms (Axioms ( 公理公理 ), postulates (), postulates ( 假設假設 ), hypoth), hypotheses (eses ( 假說假說 ), premises (), premises ( 基本假設基本假設 ))

- Assumptions (often - Assumptions (often ununprovenproven) defining the ) defining the structuresstructures about which we are reasoning. about which we are reasoning.

• Rules of inference (Rules of inference ( 推論推論 ))

- Patterns of - Patterns of logically valid deductionslogically valid deductions from from hypotheses to conclusions. hypotheses to conclusions.

Details of AssumptionDetails of Assumption

• Axioms (Axioms ( 公理公理 )) are statements that are take are statements that are taken as true that serve to n as true that serve to definedefine the very the very structustructuresres we are reasoning about. we are reasoning about.

• Postulates (Postulates ( 假設假設 )) are assumptions stating t are assumptions stating that some hat some characteristiccharacteristic of an applied mathe of an applied mathematical model corresponds to some truth abmatical model corresponds to some truth about the real world or some situation being out the real world or some situation being mmodeledodeled..

Details of Assumption Details of Assumption (cont’d)(cont’d)

• Hypotheses (Hypotheses ( 假說假說 ) ) are statements that are tare statements that are temporarily adopted as being true, for purpoemporarily adopted as being true, for purposes of proving the ses of proving the consequencesconsequences they would they would have if they were true, leading to possible fahave if they were true, leading to possible falsification of the hypothesis if a predicted clsification of the hypothesis if a predicted consequence turns out to be false. onsequence turns out to be false.

- Usually a hypothesis is offered as a way of - Usually a hypothesis is offered as a way of explaining a known explaining a known consequenceconsequence. .

* Hypotheses are * Hypotheses are notnot necessarily necessarily believed or believed or disbelieveddisbelieved except in the context of the scen except in the context of the scenario being explored.ario being explored.

Details of Assumption Details of Assumption (cont’d)(cont’d)

• PremisesPremises (( 基本假設基本假設 )) are statements that th are statements that the reasoner e reasoner adopts as trueadopts as true so he can see what so he can see what the consequences would be.the consequences would be.

* Like hypotheses, they * Like hypotheses, they may or may not bemay or may not be bbelievedelieved..

More Proof TerminologyMore Proof Terminology• Lemma (Lemma ( 輔助定理輔助定理 )) - A minor theorem used - A minor theorem used

as a as a stepping-stonestepping-stone to proving a major theorem. to proving a major theorem.

• Corollary (Corollary ( 推論推論 )) - A minor theorem proved a - A minor theorem proved as an s an easy consequenceeasy consequence of a major theorem. of a major theorem.

• Conjecture (Conjecture ( 推測推測 )) - A statement whose truth - A statement whose truth value has value has notnot been proven. (A conjecture may been proven. (A conjecture may be widely believed to be true, regardless.)be widely believed to be true, regardless.)

• Theory (Theory ( 理論理論 )) – The – The set of all theoremsset of all theorems that c that can be proven from a given set of axioms.an be proven from a given set of axioms.

… … more …more …

• Conjectures (Conjectures ( 推測推測 )) are statements that are pr are statements that are proposed to be oposed to be logical consequenceslogical consequences of other stat of other statements, and that may be strongly believed to bements, and that may be strongly believed to be such, but that have e such, but that have notnot yet been proven. yet been proven.

• Lemmas (Lemmas ( 輔助定理輔助定理 )) are relatively uninteresti are relatively uninteresting statements that are proved on the way (basng statements that are proved on the way (base) to proving an actual theorem of interest.e) to proving an actual theorem of interest.

Graphical VisualizationGraphical Visualization

Various TheoremsVarious TheoremsThe AxiomsThe Axioms

of the Theoryof the Theory

A Particular TheoryA Particular Theory

A proofA proof

Inference Rules - General Inference Rules - General FormForm

• Inference RuleInference Rule

- Pattern establishing that if we know that a set - Pattern establishing that if we know that a set of of antecedentantecedent statements of certain forms are statements of certain forms are all true, then a certain related all true, then a certain related consequentconsequent statement is true. statement is true.

• antecedent 1antecedent 1 antecedent 2 … antecedent 2 … consequent consequent ““” ” meansmeans “therefore” “therefore”

Inference Rules & Inference Rules & ImplicationsImplications

• Each Each logical inference rulelogical inference rule corresponds to an corresponds to an implicationimplication that is a that is a tautologytautology..

• antecedent 1 antecedent 1 Inference ruleInference rule antecedent 2 … antecedent 2 … consequentconsequent

• Corresponding tautology: Corresponding tautology: ((((ante. 1ante. 1) ) ( (ante. 2ante. 2) ) …) …) consequentconsequent

Some Inference RulesSome Inference Rules

• p p Rule of Rule of AdditionAddition ppqq

• ppq q Rule of Rule of SimplificationSimplification pp

• p p Rule of Rule of ConjunctionConjunction qq ppqq

Modus Ponens & TollensModus Ponens & Tollens

• pp Rule of Rule of modus ponens modus ponens 斷言法斷言法ppqq (a.k.a. (a.k.a. law of detachmentlaw of detachment))qq

• qqppqq Rule of Rule of modus tollens modus tollens 否定法否定法 pp

“the mode of affirming”

“the mode of denying”

a.k.a. “also known as”

Syllogism Inference RulesSyllogism Inference Rules

• ppqq Rule of Rule of hypotheticalhypothetical

qqr r syllogism syllogism 三段論三段論 pprr

• p p qq Rule of Rule of disjunctivedisjunctive p p syllogismsyllogism qq

Aristotle(ca. 384-322 B.C.)

Formal ProofsFormal Proofs

• A formal proof of a A formal proof of a conclusionconclusion CC, given , given premipremises ses pp11, , pp22,…,…,,ppnn consists of a sequence of consists of a sequence of stepssteps, ,

each of which applies some each of which applies some inference ruleinference rule to p to premises or to previously-proven statements (as remises or to previously-proven statements (as antecedentsantecedents) to yield a new true statement (the ) to yield a new true statement (the consequentconsequent).).

• A proof demonstrates that A proof demonstrates that ifif the the premisespremises are t are true, rue, thenthen the the conclusionconclusion is true is true..

Formal Proof ExampleFormal Proof Example

• Suppose we have the following Suppose we have the following premisespremises::“It is not sunny and it is cold.”“It is not sunny and it is cold.”“We will swim only if it is sunny.”“We will swim only if it is sunny.”“If we do not swim, then we will canoe.”“If we do not swim, then we will canoe.”“If we canoe, then we will be home early.”“If we canoe, then we will be home early.”

• Given these premises, prove the theoremGiven these premises, prove the theorem“We will be home early”“We will be home early” using using inference inference rulesrules..

Proof Example Proof Example (cont’d)(cont’d)

• Let us adopt the following abbreviations:Let us adopt the following abbreviations:

sunny = sunny = “It is sunny”“It is sunny”; cold = ; cold = “It is cold”“It is cold”; ; swim = swim = “We will swim”“We will swim”; canoe = ; canoe = “We will “We will canoe”canoe”; early = ; early = “We will be home early”“We will be home early”..

• Then, the premises can be written as:Then, the premises can be written as:

(1) (1) sunnysunny coldcold

(2) (2) swim swim sunnysunny

(3) (3) swim swim canoecanoe

(4) (4) canoe canoe earlyearly

Proof Example Proof Example (cont’d)(cont’d)

StepStep Proved byProved by1. 1. sunnysunny coldcold Premise #1.Premise #1.2. 2. sunnysunny Simplification of 1.Simplification of 1.3. 3. swimswimsunnysunny Premise #2.Premise #2.4. 4. swimswim Modus tollens on 2,3.Modus tollens on 2,3.5. 5. swimswimcanoecanoe Premise #3.Premise #3.6. 6. canoecanoe Modus ponens on 4,5.Modus ponens on 4,5.7. 7. canoecanoeearlyearly Premise #4.Premise #4.8. 8. earlyearly Modus ponens on 6,7.Modus ponens on 6,7.

Inference Rules for Inference Rules for QuantifiersQuantifiers

xx PP((xx))PP((oo)) (substitute (substitute anyany object object oo))

• PP((gg)) (for (for gg a a generalgeneral element of u.d.)element of u.d.)xx PP((xx))

xx PP((xx))PP((cc)) (substitute a (substitute a newnew constantconstant cc))

• PP((oo) ) (substitute any extant object (substitute any extant object oo) ) xx PP((xx))

Common FallaciesCommon Fallacies• A A fallacyfallacy is an inference rule or other proof is an inference rule or other proof

method that is method that is not logically validnot logically valid..

- May yield a false conclusion!- May yield a false conclusion!

• Fallacy of affirmingFallacy of affirming the conclusion:the conclusion:

“ “ppqq is true, and is true, and qq is true, so is true, so pp must be true.” ? must be true.” ?

(No, because (No, because FFTT is true.) is true.)

• Fallacy of denying the hypothesis:Fallacy of denying the hypothesis:

“ “ppqq is true, and is true, and pp is false, so is false, so qq must be false.”? must be false.”?

(No, again because (No, again because FFTT is true.) is true.)

Circular ReasoningCircular Reasoning• The The fallacyfallacy of ( of (explicitlyexplicitly or or implicitlyimplicitly) )

assuming the very statement you are trying to assuming the very statement you are trying to prove in prove in the course of its proofthe course of its proof. .

Example: Prove that an integer Example: Prove that an integer nn is even, if is even, if nn22 is even. is even.

Proof: “Assume Proof: “Assume nn22 is even. Then is even. Then nn22 = 2 = 2kk for some for some integer integer kk. Dividing both sides by . Dividing both sides by nn gives gives n n = = (2(2kk)/)/n n = 2(= 2(kk//nn). So there is an integer ). So there is an integer jj (namely (namely kk//nn) such that ) such that n n = 2= 2jj. Therefore . Therefore nn is even.” is even.”

Begs the question: How do you show that j = k/n = n/2 is an integer,

without first assuming n is even ???

A Correct ProofA Correct Proof

We know that We know that nn must be either odd or even. must be either odd or even.

If If nn were odd, then were odd, then nn22 would be odd, since an odd nu would be odd, since an odd number times an odd number is always an odd number.mber times an odd number is always an odd number.

Since Since nn22 is even, it is not odd, since no even number is even, it is not odd, since no even number is also an odd number.is also an odd number.

Thus, by Thus, by modus tollensmodus tollens, , nn is not odd either. is not odd either.

Thus, by Thus, by disjunctive syllogismdisjunctive syllogism, , nn must be even. must be even. ■■

This proof is correct, but not quite complete,since we used several lemmas without provingthem. Can you identify what they are?

A More Verbose VersionA More Verbose VersionSuppose Suppose nn22 is even is even 2|2|nn22 nn2 2 mod 2 = 0.mod 2 = 0.

Of course Of course nn mod 2 is either 0 or 1. mod 2 is either 0 or 1.

If it’s 1, then If it’s 1, then nn1 (mod 2), so 1 (mod 2), so nn221 (mod 2), 1 (mod 2), using the using the theorem that if theorem that if aab b (mod(mod m m) and) and c cd d (mod(mod m m) then) then ac acbd bd (mod (mod mm), with ), with aa==cc==n n and and bb==dd=1.=1.

Now Now nn221 (mod 2) implies that 1 (mod 2) implies that nn22 mod 2 = 1. mod 2 = 1.

So So by the hypothetical syllogism ruleby the hypothetical syllogism rule, (, (n n mod 2 = 1) imod 2 = 1) implies (mplies (nn22 mod 2 = 1). Since mod 2 = 1). Since nn22 mod 2 = 0 mod 2 = 0 1, 1, by by momodus tollensdus tollens we know that we know that nn mod 2 mod 2 1. 1.

So So by disjunctive syllogismby disjunctive syllogism we have that we have that nn mod 2 = 0 mod 2 = 0

2|2|n n nn is even. is even. Uses some number theory we haven’t defined yet.

Divides, Factor, MultipleDivides, Factor, Multiple

• Let Let aa,,bbZZ with with aa0.0.

• Def.:Def.: aa||bb “ “aa dividesdivides bb” :” : ( (ccZZ:: b=acb=ac))“There is an integer “There is an integer cc such that such that cc times times a a equalequals s b.b.””

E.g., 3E.g., 312 12 TrueTrue, but 3, but 37 7 FalseFalse..

• Iff Iff aa divides divides bb, then we say , then we say aa is a is a factorfactor or a or a didivisorvisor of of bb, and , and bb is a is a multiplemultiple of of aa..

E.g., “E.g., “bb is even” : is even” :≡ 2|≡ 2|bb. Is 0 even? Is −4?. Is 0 even? Is −4?

Proof Methods for Proof Methods for ImplicationsImplications

For proving implications For proving implications ppqq, we have:, we have:

• Direct proofDirect proof: Assume : Assume pp is true, and prove is true, and prove qq..

• Indirect proofIndirect proof: Assume : Assume qq, and prove , and prove pp..

• Vacuous proofVacuous proof: Prove : Prove pp by itself. by itself.

• Trivial proofTrivial proof: Prove : Prove qq by itself. by itself.

• Proof by casesProof by cases: : Show Show pp((aa bb), and (), and (aaqq) and () and (bbqq).).

Direct Proof ExampleDirect Proof ExampleDefinition:Definition: An integer An integer nn is called is called oddodd iff iff n n = 2= 2kk++

1 for some integer 1 for some integer kk; ; nn is is eveneven iff iff n n = 2= 2kk for so for some me kk..

Axiom:Axiom: Every integer is either odd or even. Every integer is either odd or even.Theorem:Theorem: (For all numbers (For all numbers nn) If ) If nn is an odd inte is an odd inte

ger, then ger, then nn22 is an odd integer. is an odd integer.Proof:Proof: If If nn is odd, then is odd, then nn = 2 = 2kk+1 for some intege+1 for some intege

r r kk. Thus, . Thus, nn22 = (2 = (2kk+1)+1)22 = 4 = 4kk22 + 4 + 4kk + 1 = 2(2 + 1 = 2(2kk22 + 2+ 2kk) + 1. Therefore ) + 1. Therefore nn22 is of the form 2 is of the form 2jj + 1 + 1 (with (with jj the integer 2 the integer 2kk22 + 2 + 2kk),),

thus thus nn22 is odd. is odd. □□

Indirect Proof ExampleIndirect Proof Example

Theorem:Theorem: (For all integers (For all integers nn) ) If 3If 3nn+2 is odd, then +2 is odd, then nn is odd. is odd.

Proof:Proof: Suppose that the conclusion is false, Suppose that the conclusion is false, i.e.i.e., , that that nn is even. Then is even. Then n n = 2= 2kk for some integer for some integer kk. . Then 3Then 3nn+2 = 3(2+2 = 3(2kk)+2 = 6)+2 = 6kk+2 = 2(3+2 = 2(3kk+1). Thus +1). Thus 33nn+2 is even, because it equals 2+2 is even, because it equals 2jj for integer for integer jj = 3= 3kk+1. So 3+1. So 3nn+2 is not odd. We have shown t+2 is not odd. We have shown that hat ¬(¬(nn is odd)→¬(3 is odd)→¬(3nn+2 is odd),+2 is odd),

thus its contrapositive (3thus its contrapositive (3nn+2 is odd) → (+2 is odd) → (nn is od is odd) is also true. □d) is also true. □

Vacuous Proof ExampleVacuous Proof Example

Theorem:Theorem: (For all (For all nn) If ) If nn is both odd and even, is both odd and even, then then nn22 = = nn + + nn..

Proof: Proof: The statement “The statement “nn is both odd and even” is is both odd and even” is necessarily false, since no number can be both necessarily false, since no number can be both odd and even.odd and even.

So, the theorem is vacuously true. So, the theorem is vacuously true. □□

Trivial Proof ExampleTrivial Proof Example

Theorem:Theorem: (For integers (For integers nn) If ) If nn is the sum of is the sum of two prime numberstwo prime numbers, then either , then either nn is odd or is odd or nn is is even.even.

Proof:Proof: AnyAny integer integer nn is either odd or even. So is either odd or even. So the conclusion of the implication is true the conclusion of the implication is true regardless of the truth of the antecedent.regardless of the truth of the antecedent.

Thus the implication is true trivially. □Thus the implication is true trivially. □

Proof by ContradictionProof by Contradiction

• A method for proving A method for proving pp..

1. 1. Assume Assume pp, and prove , and prove both both qq and and qq for for some proposition some proposition qq..

2. Thus 2. Thus pp ( (qq qq))..

3. (3. (qq qq) is a trivial ) is a trivial contradictioncontradiction, equal to , equal to FF..

4. Thus 4. Thus ppFF, which is , which is only true if only true if pp==FF . .

5. Thus 5. Thus pp is true. is true.

Proof by Contradiction Proof by Contradiction ExampleExample

Theorem:Theorem: 2 20.50.5 is irrational. is irrational.

Proof:Proof: Assume 2 Assume 21/21/2 were rational. This means th were rational. This means there are integers ere are integers ii, , jj with no common divisors s with no common divisors such that 2uch that 21/21/2 = = ii//jj. Squaring both sides, 2 = . Squaring both sides, 2 = ii22//jj22,, so 2 so 2jj22 = = ii22. So . So ii22 is even; thus is even; thus ii is even. Let is even. Let i i = 2= 2kk. So 2. So 2jj22 = (2 = (2kk))22 = 4 = 4kk22. Dividing both sides . Dividing both sides by 2, by 2, jj22 = 2 = 2kk22. Thus . Thus jj22 is even, so is even, so jj is even. Bu is even. But then t then ii and and jj have a common divisor, namely 2, have a common divisor, namely 2, so we have a contradiction. so we have a contradiction. □□

Review: Proof Methods So Review: Proof Methods So FarFar

• DirectDirect, , indirectindirect, , vacuousvacuous, and , and trivialtrivial proofs of proofs of statements of the form statements of the form ppqq..

• Proof by contradictionProof by contradiction of any statement. of any statement.

• Next: Next: ConstructiveConstructive and and nonconstructivenonconstructive existeexistence proofsnce proofs..