3 The Vicky Pollard Proof Technique Prove that when n is even n2 is even.Assume n is 0, then n2 is 0, and that is evenAssume n is 2, then n2 is 4, and that is evenAssume n is 4, then n2 is 16, and that is evenAssume n is 6, then n2 is 36, and that is evenWhat’s wrongwith that?Assume n is -2, then n2 is 4 , and that is even also!Therefore when n is even n2 is even!
10 Direct ProofThe square of an even number is even(1) assume even(n)(2) n = 2k(3) n2 = 4k2 = 2(2k2) which is evenQ.E.DQuod erat demonstrandumThat which was to be proved
11 Indirect Proof(0) show that contrapositive is true(1) assume that q is false(2) userules of inferencetheorems already provedto show p is false(4) “Since the negation of the conclusionof the implication (¬q) implies that thehypothesis is false (¬p), the original implicationis true”
12 Indirect ProofIf n is an integer and 3n + 2 is even then n is even(1) assume odd(n)(2) n = 2k + 1(3) 3n + 2= 3(2k + 1) + 2= 6k= 6k + 5= 6k= 2(3k + 2) + 1which is oddQED
13 Indirect ProofIf n2 is even then n is evenassume odd(n)n = 2k + 1n2 = (2k + 1)2= 4k2 + 4k + 1= 2(2k2 + 2) + 1which is oddQED
14 Could we prove this directly? If n2 is even then n is evenassume even(n2)n2 = 2kn = …QEDQ: Why use an indirect proof?A: It might be the easy option
15 If n2 is even then n is even Direct Proof againWhat’s wrong with this?If n2 is even then n is evenSuppose that n2 is even.Then 2k = n2 for some integer k.Let n = 2m for some integer mTherefore n is evenQEDWhere did we get “n = 2m” from?This is circular reasoning, assuming true what we have to prove!
16 Theorem: If n2 is even then n is even Indirect ProofIt’s an argument. Present it wellTheorem: If n2 is even then n is evenProof:We prove this indirectly, i.e. we show that if a number is odd thenwhen we square it we get an odd result. Assume n is odd, and canbe represented as 2k + 1, where k is an integer. Then n squaredis 4k2 + 4k + 1, and we can express that as 2(2k2 + 2k) + 1, and this isan even number plus 1, i.e. an odd number. Therefore, when n2 iseven n is even QED
17 The proof is in 2 parts!! Proving if and only if To prove p q prove q pThe proof is in 2 parts!!
18 n is odd if and only if n2 is odd Proving if and only ifn is odd if and only if n2 is oddTo prove p qprove p q & prove q pprove odd(n) odd(n2)if n is odd then n = 2k + 1n2 = 4k2 + 4k + 1, which is 2(2k2 + 2k) + 1and this is an odd numberprove odd(n2) odd(n)use an indirect proof, i.e. even(n) even(n2)we have already proved this (slide 3)Since we have shown that p q & q p we haveshown that the theorem is true
19 Trivial Proofp implies q AND we are told q is truetrue true is true and false true is also truethen it is trivially true
21 Proof by Contradiction Assume the negation of the proposition to be provedand derive a contradictionTo prove P implies Q, (P Q)assume both P and not Q (P Q)remember the truth table for implication?This is the only entry that is false.derive a contradiction (i.e. assumption must be false)Assume the negation of what you want to proveand show that this assumption is untenable.
23 Proof by Contradiction If 3n + 2 is odd then n is odd assume odd(3n + 2) and even(n)even(n) therefore n = 2 k3n + 2 = 3(2k) + 26k + 2 = 2(3k + 1)2(3k + 1) is eventherefore even(3n + 2)this is a contradictiontherefore our assumption is wrongn must be oddQED
24 Proof by Contradiction (properly) If 3n + 2 is odd then n is odd Theorem: If 3n+2 is odd then n is odd.Proof:We use a proof by contradiction. Assume that 3n+2 is odd andn is even. Then we can express n as 2k, where k is an integer.Therefore 3n+2 is then 6k+2, i.e. 2(3k+1), and this is aneven number. This contradicts our assumptions, consequentlyn must be odd. Therefore when 3n+2 is odd, n is odd. QEDassume odd(3n + 2) and even(n)even(n) therefore n = 2 k3n + 2 = 3(2k) + 26k + 2 = 2(3k + 1)2(3k + 1) is eventherefore even(3n + 2)this is a contradictiontherefore our assumption is wrongn must be oddQED
25 Proof by contradiction that P is true Assume P is false and show that is absurd
26 Proof by Contradiction The square root of 2 is irrational A brief introduction to the proofto be rational a number can be expressed asx = a/ba and b must be relative primeotherwise there is some number that divides a and bto be irrational, we cannot express x as a/b2 is irrational 2 a/bTo prove this we will assume 2 a/b and derive a contradictionAn example of a larger, more subtle proof
27 Proof by Contradiction The square root of 2 is irrational assume 2 is rational (and show this leads to a contradiction)2 = a/ba and b are integersrelativePrime(a,b) i.e. gcd(a,b) = 12 = (a2)/(b2)2b2 = a2even(a2)we have already provedeven(n2) even(n)even(a)a = 2c2b2 = a2 = 4c2b2 = 2c2even(b)but gcd(a,b) = 1a and b cannot both be evenOur assumption must be false, and root 2 is irrationalQED
29 find a set of propositions P1, P2, …, Pn (P1 or P2 or … or Pn) Q Proof by CasesTo prove thisKnow thatTo prove P Qfind a set of propositions P1, P2, …, Pn(P1 or P2 or … or Pn) QproveP1 -> Q andP2 -> Q and… andPn -> QWe look exhaustively for all cases and prove each oneRule of inference, p177
30 Proof by CasesFactoid: the 4-colour theorem had > 1000 cases
31 Proof by CasesFor every non-zero integer x, x2 is greater than zeroThere are 2 cases to consider,x > 0x < 0x > 0 then clearly x2 is greater than zerothe product of two negative integers is positiveconsequently x2 is again greater than zeroQED
32 Proof by CasesThe square of an integer, not divisible by 5 ,leaves a remainder of 1 or 4 when divided by 5There are 4 cases to considern = 5k + 1n2 = 25k2 + 10k + 1 = 5(5k2 + 2k) + 1n = 5k + 2n2 = 25k2 + 20k + 4 = 5(5k2 + 4k) + 4n = 5k + 3n2 = 25k2 + 30k + 9 = 5(5k2 + 6k + 1) + 4n = 5k + 4n2 = 25k2 + 40k + 16 = 5(5k2 + 8k + 3) + 1the remainders are 1 or 4QED
33 Vacuous ProofWhen P is false P implies Q is trueIf we can prove P is false we are done!prove P(3)if 3 > 4 then …if false then …Since the hypothesis in this implication is falsethe implication is vacuously trueQED
34 Number nine Disprove the assertion “All odd numbers are prime” Existence ProofProve, or disprove something, by presenting an instance (a witness).This can be done byproducing an actual instanceshowing how to construct an instanceshowing it would be absurd if an instance did not existDisprove the assertion “All odd numbers are prime”Number nine
35 Is n2 - n + 41 prime when n is positive? Existence ProofIs n2 - n + 41 prime when n is positive?let n = 41n2 - n + 41 == 41.41which is compositetherefore n2 - n + 41 is not always primeQED
36 Existence ProofShow that there are n consecutive composite integers for any +ve nWhat does that mean?for example, let n = 5consider the following sequence of 5 numbers722 divisible by 2723 divisible by 3724 divisible by 4725 divisible by 5726 divisible by 6the above consecutive numbers are all composite
37 x + n = (n + 1) + (n + 1)! = (n + 1)(1 + (n + 1)!/(n + 1)) Existence ProofShow that there are n consecutive composite integers for any +ve nlet x = (n + 1)! + 1x = … n.(n+1) + 1x + 1 = 2 + (n + 1)! = 2(1 + (n + 1)!/2)x + 2 = 3 + (n + 1)! = 3(1 + (n + 1)!/3)x + 3 = 4 + (n + 1)! = 4(1 + (n + 1)!/4)…x + n = (n + 1) + (n + 1)! = (n + 1)(1 + (n + 1)!/(n + 1))We have constructed n consecutive composite integersQED
38 Existence ProofAre there an infinite number of primes?Reformulate this as“For any n, is there a prime greater than n?”compute a new number x = n! + 1x = ( …n-1.n) + 1x is not divisible by any number in the range 2 to nwe always get remainder 1the FTA states x is a product of primesx has a prime divisorx’s smallest prime divisor is greater than nConsequently for any n there is a prime greater than n
39 Fallacies Bad proofs: Rosen 1.5 page 69 Fallacy of affirming the conclusionFallacy of denying the hypothesis
41 The fallacy of affirming the consequent FallaciesExamplesThe fallacy of affirming the consequentIf the butler did it he has blood on his handsThe butler has blood on his handsTherefore the butler did it!This is NOT a tautology, not a rule of inference!
42 The fallacy of affirming the consequent FallaciesExamplesThe fallacy of affirming the consequentIf the butler did it he has blood on his handsThe butler has blood on his handsTherefore the butler did it!I told you!
43 The fallacy of denying the antecedent FallaciesExamplesThe fallacy of denying the antecedentIf the butler is nervous, he did it!The butler is really relaxed and calm.Therefore, the butler did not do it.This is NOT a tautology, not a rule of inference!
44 The fallacy of denying the antecedent FallaciesExamplesThe fallacy of denying the antecedentIf the butler is nervous, he did it!The butler is really relaxed and calm.Therefore, the butler did not do it.You see, I told you!
45 Begging the question Or Circular reasoning FallaciesExamplesBegging the questionOrCircular reasoningWe use the truth of a statement being proved in the proof itself!Ted: God must exist.Dougal: How do you know that then Ted?Ted: It says so in the bible Dougal.Dougal: Ted. Why should I believe the bible Ted?Ted: Dougal, God wrote the bible.
46 Begging the question Or Circular reasoning FallaciesExamplesBegging the questionOrCircular reasoningTed: God must exist.Dougal: How do you know that then Ted?Ted: It says so in the bible Dougal.Dougal: Ted. Why should I believe the bible Ted?Ted: Dougal, God wrote the bible.
49 proof by contradiction proof by cases vacuous proof existence proof Proof techniquesrules of inferencefallaciesdirect proofindirect proofif and only iftrivial proofproof by contradictionproof by casesvacuous proofexistence proof