1.1 Propositional Logic 1.2 Propositional - MBA

108 Pages·2011·6.95 MB·English

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£2 Proysional
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413 Predicres and
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14 Nested
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The Foundations:
Logic and Proofs

Tica tc etistestyteamarne otter stm insane tees
vel aver pa areas wih leery sl hte ead ia
the sr UE Is qe and Hor every pubes oeyer nh sn the ee ive faegere
ot enveeding ni nbn ~ T° Laie the tis ual albert rewebain, ea a
coma rescning thes prelical piston inthe csi af empang mashines, oe
‘joe Fcatian nf egricms, ts afi intligeres, ta ecm eran, peegraag
Tnguapes, see het aren fonempmer nes, a wel a a mary oe eso ss.

‘Trnestand maths, ws Misr undead one make 9a enroet mas eas
argument that is apzeat Once we provoa muthsmarca seta stra. scecl te tore. A
fofeesion afeneencanatopisamgenize white rv ahoutthistope, fo lem ezsathemetica
fic, porn needs en astivey eonenuct mathematical angamenss or Bi ee, and ot jst
zand Sxositon. Moreeser, because knovsne ths rue of a ears ofion rakes possible
‘emily the seul o Hien sitions povfs lay au eet role inthe devecprnen! uf
row ides, Seas of exmpeter sence len find it sung be impentant prot at i
fumnpuler scien. Ziel, evel play essen zo: whe we Yeti tt wouter pryrans
pial the gees orp Trl puss clin, ben ae abe thal aarti they
Draw the suet real, when we ect the stiy a oiem, aa chen ae eae

fatieligenc aged vein ystems ave bewn cans hate nau:
te cone shir ajes,

Tdhis caste, we wil ceplin wat rakes yp onrret mahemstiealaygumen ane ins
scenes msonatiogt hese agus, We wi devs anareenal of dire ron mths
thats nae us prove many 4 ferent eypes of teen. Afar intndziag many diners
mmetheds of rect. will nmoxuoe sores stetegy Fe ens uting prot. Ae wi nroduce
fhe necion oF a coviccmre and expan the prucess oF developmg Teutemates hy radstnyg
confess.

L.1_Propasitional Logic

Introduetion

The Tule of logic vive presse mesving “a mathmusial stutomems. These coin ane wed £9
sistinguist bers validand nei sntheratical argues. Becas: zor gl hs Book
‘330 leach se eager fo lo urderand anal how lo cumius cercel wahenaaal wpuszens,
‘we hegan our study of userets muthoratcs wb afro te lage.

Iadgiven cw i impurtante fa undersanding ateacal rensonin. yin nurs
applica in corguterseienve, these rales ae sed inthe design of Compare cia, Le
‘eonstucion 9° oumputerprogeas te veiicson of De sousomess oF programs, andi
iouy ole: ways hurieceurs, sullwale setete been devéuzed forensic pues
avtomgically Sil gisenss Rose appicaions oop nae ppsnaning chara.

Propositions
Ou davussion eves wit an asada ote basic buhlina Banks tlie preston,
A proposition dtl secu (ie see a lave Fee esi rue
te ise. bus ws bi
2-1) The Ramos Loge aa Pt 71

EXAMPLE 1 Ul he flowing Uecbirative sentences ae arposilious.

1. Waskingtun, DC. isthe capital ofthe intel Stes oP America
2. Toromte:s the cull of Canada

Blt 2

4212

Proposilons 1 and are trae, whereas 2 um 4a fie “
Some ventenees tut ae nt positions are given in Example 2.

EKAMELE2 Coasidor tho tllowing semenoes,

4, Wt ei i
2, Roa his caret:
Baxtl

exe

Sentonecs 1 and? are nar propasicans hecauss thoy are no: declarative sonronesa. entenes 3
and 4 ave norpropeations hecaosedxy ars ether ue nor false, Notethat ek afecrmencee 3
and 4cap be turned into a propesiin if we stsgn values othe varases, Ws wil alse dias
‘ter aap ota searencee sich as these ate propositions in Sect 3.2 “

‘WursTener ta denote propostional variables or statement variables), has, anaes
that epresnt propasiions. just a eters ace used ts denote numerical vanables Pe vooven-
tionel eters used for prozositionl variables rep. q.7.1..... Thetruth value oft propoation
ieares denoted be", itis atm proposition and fas, denoted by & itiaw fase propos.

The areuof logit deal with proposiuos scale t= proposition aeulus or prope
sitonal ogi, II sis Bit develapet systems’ by the Greek philowpher Astlle mre
tun 2300 years ago

SRIGTOTIE 84a 30 cry ae ham Sa (Shin mem Gee he er
$e hs veer nel ean dan ga ot ie we ee
“ehrefor 26 ycnsheaizided Pe's ea ft aresctay ug we baz tri, Whi Pa dodo
‘nan Atae van cen nee cr bectae eee ea etch Goose of Pe
ite Seo otal the Spa ing Howat be arcu seca ad asd wee
the Se Wena Roma dl Heuer, aco eda tn a
‘ef tasdoa, bo sd Alcan Pulp con ho tr toast he Cie ots erat Arnie fr esa
Stator the dca of tea Yul, bo ented co Acs sad set bis oo sch ae tb cer

“Allies floss wee clos bs erbea, ic Uma "0 tlk aon tate cle oe walked unt 35 he
Aeconand cept potion Ares mg a th sco er ye whee bee a ftanend ses he
‘ie i land tpn antes asin Ae. at ata ann
ce Tc er eed ce hy hy ag ennai Aa we
TD, Segoe er am erin wo metas fx prodecze ww papesitions fom rn or

tinalbad se already hae, These metbods wete discussed by the Falish zetheratician George Boate
in 1s in hit hock The Fase vf The May nematic’ sttenants ae cenanet Hy
Combining one nr mare propasicans. New propasitinns, called eampauna prapestions, are
fered trom exiting qropas ans using logis operat

DEFINETIONT Tat p'sea proposition. The mutton of, denoted by “paleo donowea by Fh sthetacmnert

"ic sot he cane at

‘Tae preposition —p es oad “not p.” The uh alse oF
ofthe truth value a

reeation of p,—p. isthe oppose

EXAMPLES Find th noguian of the prapocion
“ody is ia

cot
Cenntes Mad and exgrosehisin simp English

Salucons The asgation is
1 ig arte ee hat rosy i Fig”

‘os negation can be anoce mpl expend by

“Taday i wor ey?"

Thi at Fry ti” «
EXAMPLES ial the seyaton uf te propusitiog
“As eat 30 hes orn tl soa in Miah”
aed express this simple Fgh
Satine The negation is
“iniunot the eave tha at oust inches ri lta in Mia”
“This megaton ean be me simply explested by
Le than 10 inches ef xn fell yn Mai” <
Remar Suit sraking, srtencs incaing ariahte mes such as nse in Fxamplos3.and
FABER] The | dprenge propos tango esa feed ime is assume. The same tlds oe varible places anleas
Bre BEB | aoe pce assumed and for omouns wast parce eon i sumed: We wl abs
Fropafon, * | Se Seed ies. red pare end peru pep nck semis ws ee
TP Table 1 displays the truth table forthe negaton of a proposition p. This ble hus uo
F | 7 | foreach of the tno pusisle uth vues of propoalaa p. ah ro sows hl wae of
=p correspancing to the eth value of pie this 70%:
41TH Feta: Li al Pou Mo

DEFINITION 2

eXAMPLES:

DELINITION a

Tie seyaion of progesiton can also be cqnsered the reat 9" he operon oF the
negaton eperstor 4 propoiion. “he ngayon aporateeconstracs a new proposition toma
A shleettting opesiian We wil nen intone the apical apcates ata ied Yar
hes prog fm tamer move evstingneonnsiins. The nga aerarnsae al sll
cammectves

1Let pang be propositions, The confuncion ofp wal ylenuby p A gis the propesition
“pind g? The voajuncton p 44 lve when bois pnd ae ue aa is alse oer,

“Tie 2 dispags de tala els of 7% g. Thin te has a row Fr eas of the fr pie
soivhinaions OF Mth aes of 4nd Ths far oes correspond ea dhe par of tut vahast
‘TE.TE.FT, ane FF where rhe fie ru vluo inthe par she ath wal ofp ané the second
‘ruth vai fe the truth vane of

‘Note that Sogo the word “at” roracnmey is used vstead uf“und™ ia conjunction, Far
sample, the staresant "Tho sun ssbmang, but cis ening” isnaocher vy of saying be sue
{5 shrang and st isrining la tunl Langosge, bere isu subtle edfereave a mezoag oe
al" and bul; we will ol be oneeruel wth this mance ere)

Find he canunecinn ofthe reposition: p and g wher p
and is the proposition "its raiing todas:

nthe propio “Today tia”

Sotacon: Tre conjunstion of tase prposiuns, p & gis ke proposition “oun day an
Fee alng tay Ths peopostion is teu wn ety ids ald Tse ony dy hat is n0C
{eieay wal on Enda when i does tra «

Let ant he propositions. The digiestcn ot p andg, denon by p v gy th propesition
“pong” Tha dsianccon jg ie ale when Ich ad gare fale and bx mae otharwie,

Tbe 3 ups che rath fabte for pg

These ul the eonettve or ia Ui uct veresponl lo ome of the 4 ys the wont
cris yan English rarely ina incline way, A junio yee when staat ea th
‘any proposilions iste, Foritatata, einai oi ig used nthe tae.

“Siudenta ws ave taken calculus or sempaterwionce cunt th elas

TABLE2 Thera table tr "TABLE 3 Che ath tbe for
she Convection oe he Pafvestion oT
Proposons ropa.
1 agit bogs §

Here, we en hal sl en's who have rake both calculi au’ onpute selence can ake
lags we athe stacents way have lakeu iy ove of be Ie 32ers, Du ke ole ha
neste wang he excliiee or wien ay

“Smee whi have rk eal or compe seine, fat thot, ca ee in hi

ere, wt mean tal stents wha ae taken fath celebs ard a compte seonee eomise
‘amor ke the ass. Only these wha have teen exactly ne nf th twa crue can take “he
hae,

Similar, shen & menn at a reeanart sates, “Soup or salad ones with an entrée” he
rewaiaitalostatwnys cane dat er-tomers can hve either anup or aaa, fit not both
Hero; hs an elusive rather cha an inlusve

EXAMPLE 6 Wha is ihe difancton of hc sropostions panda where p and arc the sance propositions
in Examples?

Soturion: he disfution ofp und a. p 9.1 the proposition

day us day we is cain today

“Thisproposition is tue cm any da that i ether ide are rainy day etude ain day)
only false on dis but are not cays whe also does le <

As nus presuusly cemarked the ust of the comaertive a a ius comes
fo uae ofthe two ways the more orm used i Logs aime. mon incline wey. thos
lsjamion i tee hea epl te a Feet propio I se. Noretimes, Ws D6 or
In an exclusive cence When the exclnive oe fs near connect the preposition pal the
smupeilin oz (bt vel both) is ined This prpesiion i tae when p feted gis
Faben sn wr fae andi Wale hhh a are flee a wen hah ae

DEFINITION 4 Let pad be propositions. The evchuriarot panda, denotal vy p 4p ithe propesion
Xtae lun when exaely one oF pail gH eae fa is ele cheese

ul be fur be exclosve ur of ly popes pled in Tale

wa
anace nOON at Rm ts do nn nL, te
i | Site ene oi iaalps Se at don tesco amulet ch et
1 Muppets Neel stats ee
Se et ae ae a a te
Ep | escsonc te pha agen reatuncanagh boi oma ore
SESS RAERS Ste le are clots thse este er Mes
CIR eet thet erations
ts 18he manent msoraronanowaies Gaasnt eg mn wna anepalies bebe he
Toned th ec heehee anes Gs Cle et edd oc i
6 Tye Poa: Lape Ba A

DEFINITION 5

ae *)

TABLE 4 The Truth Tati for
£ he Pecanive Ore Fe
Propestions.

Conditional Statements

‘We tl sens several other inurl wa isk ppostigns ens be apmine

Legrand be proportions. The condinona sairment y+ ¢ ithe seopeition “i Thea,
(pTherendicomalsatemen p> 8 fusewhen fisemeand i fle, and rte fherase
tee conditional sarensant p> i calle! He Aypathots (or ante or promi
and g iealed the anatase or conioguen

Thesamenen gy ical a conitionsl taunt hesouse + g assets ag ims
edhe conaign thal p ads. A eal sant i alo calle! a impel

Ths erath able Yor the conditional sacme p> inshore i Table, Mote chat the
seatomene 7 > fe true when hott p and ate ric and whon pea tilas (no master what rah

a.

[Because condtional stccment ply euch an ssi! colo in muhematicl easing,
sary of terminology is uned to express pq. You will eaoouatee mist if not al of hs
fallewing ways express his condienal statment

ip. thee 9” <p ilies
sing” Sronly a

pie ubisien: org “taut condition forg ip?
pity” sp whensver p”

“a hcp” "9 isncocesay frp

‘snovorserycndtion thr ping” "9p Zllowe fom 2

‘gules: E

A oseful nay to ume the teh value of w condition stalemeat isto Mhink 9f wn
obligutionorovotcelForenumple tbe lelgeeuny pliicins mule wbeanuaning for ellice’s

“DEL am elected then T wid lover ces

be puis elected veler vale eect Dis plc lw tower lame, Feslheone ie
poliligian fs nul eles, Hew vole ila ave ny eects penson wun
Sa alow the pig ray aves FMgente by eas Hee aoe Bee as
Ic izenly when te paca is eiceed bur dae not hot tees that voters can say Ca He
Plceian hs beck the carapaigh pode. This fst acenarioentresjonde ta tho sae when
Feteur bug ie fale inp > 9.

Siniltly eansdes¢ einen sha a wofesorenghe make:

“ify get LUO on the fina en you il act an A>
EKAMPLET

yeu manage to get 100% un the fs. ie you would eapeet to eve a A, TE yu do 201
pet 160% yuu may or may out receive a A Uepeniiag us oles Laclors. However. IC you do gel
Tue, ou the professor dome aol give 70 a, yo wl Fel chee

Mang: yense fal trating thal" only ig” axgseses the Gene thing a8“ p Gen 3°
‘Ta remember thi, nak that panty ig srthal pcannnehe ame wheh q is vot ine, Th
nested Fler Ip ue Ing fle. When p i ee, 9 may Neches oF fko.
hecanes she stem says wntig abut Me anh valve ag. A samen err ie orp ea
tink har“ only if Eva way of cxpmsting p+ 9. However, dive statement have dierent
swath wanes whor and ghee diffezont uth wana.

“The werd “ures” oft meet expres enaditionalstamments. Osceve that unless
mp" meen thutie sp fle, then g must be trae. Thai the tere “9 unless pts Tse
ten p ierue an 9 fale, bat it sie tberwve, Cnsesuenty "i unless 7p" aad 2 > @
akeay have the same ath vl

We ilurrie the turaation between conditional sulsents and English setemens ia
Examp.e?

Tel p he he alemant “aris Wat teat eathanvatioe” aly Mh alee *Masia wi
Find gel uk Fens he stein ant p+ g ana loner in lish

Solution, From tho dstiniion of enndinal sttemene, we ae that when 8 the satinent
“Maria are dicts: mathomariow” and iethotatomenc” Maria il ada eee jl.” P— 9
representa te arom

“IE Maia Yarns discrete malbeantcs de she wil fud9 good job”

“These ure many olber ways ty capes this conditvnl sated in Fgish, Aowong the!
tical of ete ae

“Maria will find goed fly when she leatoe Siatete mathematica”
“Far Maat go a gos Joh it cist for er ta am eine mathomatis.”

ud
“Muri wil find good jo nes she dosent eum dere mathemati.” <
ote thi the wy we heve defined cuadidanu ateroeul is nove gene hn he mien

sauaced te sue slates Lathe Saghsh laguoge, Pol instar, le ouaetional street i
xtmgle ? mi Me slatemneat

“a isan tev ther weil gta the each

sre statzmns sod in noeoallangnsge vor thre ita rolaronship befucen the bypothesis
find the conel.sinn. Further, de Ris of these statements i tue unioss Mana (seme ducts
‘mathematics, tse Gnes not act «good jh. end de socand is rue tle iti ndocd soma
ay, hut donor yo to ths beac On tho her hand, the sate

“Today i Fide. then 2 = 5.

sve from te deliiva uf «vom sateen, bes ty vance ne. (The ah
‘aloe ofthe byputhesis doe not arth.) The anional semnent

Ty is riya 243 = 6"

ruc very day except Fria even dough 2 — 3 = 6 fae
‘We wooldnoc ue ese ust thn eikonal ferent ia maura uae essepe pecbps
in sarasmy hoeanse tore ina rlatiaashn etwecs the byphesivand the conclusion in eit
TXAMPLES

EXAMPLES

stulerent_ materiel reasualag, ve busier dion salemerts fame geen sort
{Waa we vn i English, The ahora. concept ofa condiionalestment sindooeucontor a
‘tunel ne aliodsi bles lypothesiand co ying Chl Ais af poticnal
Sint pein enh cle i ho hina Fhglish ape, Peopnstional npg
Icanifvil lngnge ely pall nga sagem nike ihengy se ase and remenhner,

"The HEahon emeraton used in any aregeanming‘aneuages ie Afiocnt fam tat used
in tegie Mar yungranining langanges eneain stem suchas p ten, whsre ink
eopnstionand 5 apmgram scament (ee ar-morestatomensstabecxcoata). When exeution
fa peegiam encounters suoh a aremert is excoute# p is mu but a at enccute
Istalee a6 FInsoare in Example.

While ya oF he sie « ls the sateen

Athens = 2-1

Ie 4 =0 ote thie tement ie encounered? (The symbol := stands for assignment. "the
statomenc = +] meaDs the welenment ofthe vakweaP x 11021

Sblution: Beconse 2— 2 = Are rue, the asugnment statements =x Lisecuted. Lecce

has the valne® + [ — [after thi satemea is encountered

COWERSE, CONTRAPOSITIVE, AND INVERSE Wie ean form soms new sondsinal
‘uzmene string with w condicana: tacement 7 — g. In pricey, thre are thee relied
conditional sztomentshatozeu ta ofon tht ty have special aris. Ihe poupasionag > 7
incall tr eanverse fp —~ q. The eontrapusitve ct p — 9 inthe propasiion =a.» —.
“Te proposition =p —> 7 in called the inverse of p-— g. We wil ee thac of there oe
ccpaltingal surements farmed frum p+ goats the cuabuposiie vas is he site ist
alae as p= @

‘Wetint sow thr the vontrapenitive,—y + ~p, fu conta semen p= a always
fay cae sume uth values 2» Tu see his, ute bul he cunuapusie false oly when
Ap ele an ~y i ae hac iy whe gue aid i lee 82 bone show that eid
fhe convene g > p. But the incre, pr) yg. fu Ie sine Ik she a6 pq fall
osble ul values ofp sal yo il whea is tee ali se the vigil conn
Sateen ie Fe il the ears and he erse ae Be es.

"Whe ty compound propecionealeas have abstr valpeweesll hemegatealent,
scoluta conde slatenearad is comaposiveaoeqnivalsa.Taccorvcrseandthoineerse
fs confional taccmort ae ale cqubvalene as th race cut verify, hut neither zauivlen!
{the original condticnal stomcnt (We ll ody equvalere propositions fa Setiva 123
Tako nate that ene af tis mae corimen logical crm assis thatthe convene or the
inverse aa conditional tomcat is squivalont 0 his conan stare:

‘We iustute the use of coaitional statements in ample

Wat ae the soligusttive, the irae, and the vera of the coin staerrent
“The home team wine whenever ie cnn”?

Seton: Because “y whenever p ix aay of the ways Wo espres He voniional vatemsat
1p ule wiginal statement wal Be reer

"Uicie mining, tho the home team win”
Conzequetly, the cotrupoenve of ths conditional stsement

“Tae home fam does nor wi, then icc or xing,
DEFINITION 6

EXAMELE 10

a
baum

The converses
“He Nome ean wing hen ei raining.”

The inverse is
“TFr isnot raining. thon tho horas team daes noc in

Only the cuninpusicve equivalent to te original stern <

BICONDITIONALS We now ntadce moter say 0 combine propositions that expresses
thal toe propusi ns Mave be sane il vale

Let p and g be proportions, The blsontinal satemant p <4 ie the pooposiion “p if
and only fg” The lncanditiona sate py is cue when pandg have th sane uth
‘aloes, ands fale otherwise. Bizondiconal aatements arp als alld biimpications.

“The eninahlefor p + shown in Tale. Note tat to satsment p< g istraswhen Bath
‘hocondtion satoricntsp > qandg ~ p arotus ands fale erheris, Thet way We UES
th wezda "and only fo expsse ds legal connective ad way cin ayotaleeywerinou
‘by combining the aymbola — and. Thor are samo thor eonnno sa to express p= @

p isncoasary and ston fe g™
pron g, and carne”
spina

“The last way of expressing the bienaditinal statement p + utes the sewn FP fox
Wanton? Nave that ps g hae ctlly he same seth value asp — Qh (@ > Ph

Te pI he statement Vou ss ake the High ga lly Do the salam "You Tuy ation”
‘Then p eg inthestaemnen

You can uke the Bight {une bay i you buy eke

“Thistaerent ise i pa gan ehot Doth rue Bol abs, yu uy aka
ak the Mig oF you ty che and yoy van lakes FEgt. Hs falas wea
Ufatdg hess ofa rth values, tas, whan yon do aot buy a eke, ur you can tke the
Figit Gh ax esa you gt fos Wp) aden yo Ing’ a her and eanat take hs Fight
(ok gt whan dhe ating Puts you 4

TABLES The Toa
Bicendtonal p =e
1 Tee Fomnains Leg w Pels ka

EXAMPLE 11

IMPLICIT USL OF BICONDIIONALS _Yoo show he svare Car Bcoaditonal ars et
alsays cxpict In naa Tanguags. Te perticoar. the “UF an any 4" conseuction ose in
Incense uso hy cameo language. Inston biconltionale ae ais cepresed
using aif the” an “only? ennesmtion The other part athe and yim
"Thas's e cowerse i inplied, hut not tee. For example, enmser the utenti Llib
“TF you fini’ your mal then you cin have deserL” Svat is ealy mene "Yeu ca bave
ester and omy i yor: Bish Your meal Ths ant sateen aga equivalent te et
stutemenes “Ifyou ish your mea), Caen jou em huve dessert “You cam ave deste! oaly
‘you nich your meal” Becunse of tis presi iu etal -autge, we teed aks 2
‘smphun whether vndticral tated! a nalal raguage imply desi oot
[Bzcuie precision ie ensmtal n mathematics und ia loge, we oil ava dias betseed
the cundianal steel ¢ + g al ie bivadltivsal sateen! py,

‘Trath Tables of Compound Propositions

“We hive now inte four important Logie eanactives—conlanetions, gimetons, ean-
itiaalsitements, al bicoadt onal statmaen—ae wel as aagetions. We can use thse ea-
gets to bil op complicared eompoan proton nsniing any sumer of postal
‘arabes, We can ns uth tablet dori the tru. alucs of hese ompond peopontions,
fs Pramlo 11 eataes, We tas a spate elo t find ta tu value each compen
sxplestnn ret seen in ho eampound propestion ai ix bat mp. Th wrath veluas a the
‘emmpnnd pronasizon tor each conmhnution of mith ales of the propiinal arabe
‘found fn the fina column oie ble

Comtarut de tru tabs of th compound propesicon
(ena qian.

Stution: Because dis mh table ineolves oxo propositional variables» an, ther ure ou
‘ose in this rut thle, coresponding 19 the eambinstions af ruta vatuce 1, LE #1, and
FF. Tae fu feo columns as used forthe truth else of pad, competvely, im tho td
columns And ths uh velus eff. acoded 1a tnd the ath value of pg. found in
the fourth colnean The tata ve 3° p Ag 1 found 2a ee fit column Finally the teh
tne of (pv 9) = {p 491s Cound in the-ant columa, Tae reiting tui lei alow in
Table 7, «

Precedence of Logical Operators

‘We can construe compound propositions using the egulen operator ane te logis vperabns
defined se i, We will penersly ee pureibests wo specify he ure ie wih logical upertces

Peli |e] 3
pee pe} 3