Proofs Cs 330 : Discrete structures a proof is ? what - - PowerPoint PPT Presentation

and Rules of Inference Proofs Cs 330 : Discrete structures

a proof is ? what - mathematical reasoning deductive reasoning - - valid argument that establishes the truth of a conjecture - systematic demonstration that if some set of assumptions ( hypotheses ) are true , then some conclusion must also be true - proofs may leverage " known fads " - axioms .

use to build proofs There are many techniques we can , no prescribed recipe for how we go about but there is coming up up a proof ! : how to solve a Jigsaw puzzle ? Good analogy

Rules of inference describe valid transformations of on tautologies logical statements based . inference reached on the basis : Kwun ) a conclusion - of evidence and reasoning what logical assertion can we make based on - some set of premises ? → q , assuming propositions p are true , p e. g. , what can we assert ? q must be true !

Rule of lnfrna syntax : } 1 premise if all are true , 2. premise conclusion # , n premise - must also be true

" ) " mode that affirms modus powers ( Latin : Rule : → g) np ) → q : ( Cp tautology → q e.g. , if the AC is on , I will be cold p the Ac is f- on of therefore , I will be add

" mode that denies " ) modus Tokens ( Latin : Rule : : ( Cp → g) req ) → n p tautology → of . if the AC is I will be add p e.g on , not cold I am ^G_ " P therefore , the AE is not on

Hypothetical syllogism Rule : → g) nCq → r ) ) → ( per ) tautology :( Cp → of I eat candy , I will be wired if p can't sleep q if wired , I I am → ✓ I eat candy , I can't sleep P , if therefore

Rule : Disjunctive syllogism : Gpa ( prod ) → q tautology will not take Econ " P I e.g. , I will either take Econ or PI Soc of therefore I will take Soc ,

Resolution Rule . . tautology :( Cpvq )nGpvrD → Cqvr ) pug x do > 20 or y -102 × 7-10 or 2- LO GV r . y > 20 or ECO . .

: Addition Rule p → Cpr g) tautology : = 4 I 2+2 ✓ 90 P a rockstar zt 2=4 I am or

Rule : simplification ( Decomposition I left : Cpr g) → p tautology M £ 115.4 ! Ying - P hypotheses

Rule : conjunction ( construction : Kp ) nlq ) ) → prof tautology P g- pig

we can also replace any logical expression Remember that an qui ( or part of a compound expression ) M valent one . . using De Morgan 's law e - g ← pvnq n q ) n r - ( p

we can also introduce known tautologies based on statements preceding . , using Disjunctive syllogism tautology ( tip nlpvql ) → g) E. g. n ( ( an b) v c ) n b ) Ya ¥ c) → c - C

A valid argument is a square of statements , where statement either : each a premise ( we can stale a premise at any time ) - is - follows from preceding on rules of inference ones based - sometimes ( the last statement is the conclusion but what we are trying to prove not always ! ) .

ftp.PIscqr ) . . premises Eg s of prove , p → s ( premise ) i. T s ( premise ) 2 . 3 . Tp ( modus tokens ) → ( gnr ) ( premise ) 4. Tp . afar ( modus powers ) 5 q ( simplification ) 6 .

E.g. , premises { I÷ ¥ fqnr ) 7 s : prone . syllogism ) n r ( Dis prof lpnemisr ) i 7 . . q }( simplification ) P s → r ( premise ) z . 8 . z . modus tokens ) . Is ( 9 → 7 ( afar ) ( pneuma ) . p 4 7 ( afar ) ( modus powers ) 5 . 6. n que r ( Demorgans )

Rules of inference for quantified statements : t xp Cx ) - universal instantiation ( ul ) . - . pas PG ) for arbitrary c - universal generalization ( UG ) : F ¥ : Fx ¥ - existential instantiation CEI ) Pk ) for some c - existential generalization ( EG ) PG ) for some c : - F x PG )

txCPCH-s@xxlnscxDJS7txcpcxsrrcxDpronestxCRCx7nsCxDl.V E. g. premises Scc ) - xcptxnrcx ) ) ( simp . ) Cpnemia ) 7 . Rcc ) ( simple ) 2. PG ) - RG ) 8 Cui ) . Canis ) Rcc ) nscc ) ) 9 3. PG ) ( simplification . . ttxlrlxhscx ) ) 10 4. tx( Rx ) → ( Qcxjnscx ) ) ) Cpnennsi ) ( UG ) Cui ) 5. Pk ) → ( accuses ) 6. QQ ask ) Cmp )

mathematical theorems are often stated using free Note : variables in its hypotheses and conclusion , and over there free variables is implied universal quantification . : if , conjecture - then I E. g. > u u > 4 - - Q cu ) Pln ) Pcu ) → Q cu ) for arbitrary i. e. , n universal generalization : we want to prove Hn ( Pla ) → Qcu ) ) p → of " form " of proof goal :

Methods of Proof of form p → of . Trivial proof : q known to be true 1 " if it is raining " then It 2=3 e. g. , z . Vacuous proof known to be false : p z > 3 then Elon musk is a genius " " if e. g. ,

Methods of Proof of form p → of . Direct proof : assume pi prone of 3 - use axioms , rules of inference , equivalences 4 . Indirect proof a) proof of the contrapositive ( recall p → of ⇒ of → a p ) n q , prone up - assume b) proof by contradiction r e q ; derive a contradiction G. g. , rn - r ) - assume p

Methods of Proof of other forms 5. proof of taconite tonal p ⇐ q prone p → q and q → p - . proof of conjunction p ng 6 - prone p and q separately . . if hypothesis is a disjunction , e. g. , Lp , v par i v Pk ) → q 7 . - → r ) n Cq → r ) - un equivalence Cpr g) → r = ( p a Cpa = Cp , → g) - Cp , v par . n ( pic → q ) . - up e) → of → g) n . . case IT separately each - - prone - .

Methods of proof involving quantifiers . proof of form tx PK ) 8 ) for arbitrary PG - show c 9 . proof of form at xp Cx ) = Fx n PG ) - find a counterexample 7 Pcc ) c where - " existence proof . proof of form Fxpcx ) " 9 " constructive : find c " where PG ) proof - no a constructive " proof - " where PG ) i no c exists : assume derive a contradiction .

' others Many . - mathematical induction - structural induction - cantor diagonalized voir - combinatorial proofs - etc .

Direct proof E. g . x is odd ( ie , we can write it as zyt For all integers x , if I , is an integer where y , then I is also odd ) . proof : - let x be an arbitrary integer - x is odd so x = Zyt I , = 2 Gy 72g ) t l = ( z y ti ) 2 = 4 y 't 4 y t I 2 - x - 2y2t2y is also an integer Z ; ie . XZ = Zz t I . he is odd ' -

E. g. , proof of tnkouditional / conjunction / cases , by contrapositive For all integers x , K is odd if and only if x is odd . proof ) n @ is odd → x is odd ) show ( x is odd → it is odd : must already proved ! I 9 handle second case - try contrapositive x is even → E is even : - if can write it as 2g x is even , we = 4y2 = 2 ( 2 ya ) = Zz 2 x - X 2 is i even - - is odd ⇒ * is odd ' X - -

. proof by contradiction E. g There are infinitely many prime numbers proof : - assume there is a finite list of primes pi , pa , . , p n . . m =p , xp ax . x put l - let . . - m is not divisible by p . ( would give quotient of pox . xp n , . remainder of l ) also not divisible by pa , . , Pu . . are either prime or a product of primes , - all integers > I - either a new prime or a product of a prime not in our list - m is - but this contradicts our assumption of a finite list of primes ! there are infinitely many primes ' - . .