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Introduction to Mathematical Thought

by: Breanne Schaden PhD

Introduction to Mathematical Thought MATH 124

Marketplace > Boise State University > Mathematics (M) > MATH 124 > Introduction to Mathematical Thought
Breanne Schaden PhD
GPA 3.72

John Bailly

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John Bailly
Class Notes
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This 36 page Class Notes was uploaded by Breanne Schaden PhD on Saturday October 3, 2015. The Class Notes belongs to MATH 124 at Boise State University taught by John Bailly in Fall. Since its upload, it has received 7 views. For similar materials see /class/217999/math-124-boise-state-university in Mathematics (M) at Boise State University.


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Date Created: 10/03/15
InTroducTion To Logic amp SeT Theory M124 Ou rline of Logic lec rure 1 In rr39oduc rion To course Tex rbooks Wha r is formal logic Pr39oposi rional logic Pr39oposi rions Pr39oposi rional connectives Formalisa rion of arguments M124 Logic and SeTs Proposi rional Logic Basic ideas proposi rions connec rives Formalising s ra remen rs in na rural language39 Formal proofs Se r Theory Basic ideas defini rions of se rs Rela rions func rions and equivalence rela rions Cardinali ry fini re coun rable and uncoun rable se rs Predica re Logic Logic programming Web Pages h r rpma rhbioses ra reeduMa rh124h rm All ma rer39ials used in course will appear on course web pages Power39poin r slides Homework shee rs Homework solu rions Please don39f print The notes I39ll hand ou r copies each week InTroducTion To logic Who r is logic Why is if useful Types of logic Pr39oposi rionol logic Pr39edico re logic WhaT is logic Logic is the beginning of wisdom not the end WhaT is logic Logic 111 The branch of philosophy concerned wi rh analysing rhe pa r rerns of reasoning by which a conclusion is drawn from a se r of premises wi rhou r reference To meaning or con rex r Callbs Eng13h Dic anary Why sTudy logic Logic is concerned wi rh Two key skills which any compu rer39 engineer or39 scien ris r should have Abs rr39oc rion For39moliso rion Why is logic important Logic is a formalisation of reasoning Logic is a formal language for deducing knowledge from a small number of explicile stated premises or hypotheses axioms facts Logic provides a formal framework for representing knowledge Logic differentiates between the structure and content of an argument Logic as formal language In This course logic will be presen red as a formal language Wi rhin Tha r formal language Knowledge can be s ra red concisely and precisely The process of reasoning from Tha r knowledge can be made rigorous What is an argument An argument is just a sequence of statements Some of these statements the premises are assumed to be true and serve as a basis for accepting another statement of the argument called the conclusion DeducTion and inference If The conclusion is jusTified based solely on The premises The process of reasoning is called deducTion If The validiTy of The conclusion is based on generaIsa on from The premises based on sTrong buT inconclusive evidence The process is called inference someTimes called inducTion This course is concerned only wiTh deducTion Two examples Deducfive argumen r quotA lexaha ra Is a oom or a holiday Pesoquot7 Alexandra is nof a port Therefore Alexandra Is a holiday resorf Inductive argumen r quotMosf sfua eh 7 s who did hof a o fhe fufora ques ohs will fail fhe exam J39 0 did hof a o fhe fufora ques ohs Therefore John will fail fhe exam Some differen r Types of logic His ror39icolly a number of Types of logic have been proposed In This course we will s rudy Pr39oposi rionol logic Boole 18151864 Pr39edico re logic Fr39ege 18481925 Proposi rionol logic Simple Types of s ro remen rs called propositions are rr39eo red as o romic building blocks for39 more complex s ro remen rs Aexana ria Is 1 pam or a hoia ay resort Aexana ria Is 1707 a pom Therefore Aexana ria Is a hoia ay resom Proposi rionol logic Basic pr39oposi rions in The argument are p Aexana r a is a pom q Aexana r a is a hoI39a ay resort In obs rr39oc r form The argument becomes P 0 Nofq Thereforeq Pr39edicaTe logic Ex rension of propositional logic A 39pr39edica re39 is jus r a pr39oper39Ty Pr39edica res define r39ela rionships be rween any number39 of en ri ries using qualifiers V for all for every El There exis rs Example LeT Px be The pr39oper39Ty 39if XiS a Triangle Then The sum of iTs inTer39nal angles is 180 In pr39edicaTe logic V xPx For39 every xsuch ThaT xis a Triangle The sum of The inTer39nal angles of XiS 180 quot AnoTher39 example Le r Px be The property 39Xis an in reger39 and x2 4 Then 3 xPx Ther39e exis rs xsuch Tha r xis an in reger39 and X2 4quot Newton39s second law of motion 660199 for every x of type called object V x Objecfo sfa onaMX v Ihunfor39mmofon X v 3 f 39 Force 0 X I39sacfeduponby f I there exists an f In English for every x of a certain type referred to as an Object x is stationary x is in uniform motion or there is an f of type F orce such that x is acted upon by f Vanda Remember V X 39for39 every 6 or39 39for39 All 6 El X39Ther39e is an X or39 39Ther39e Exis rs an 6 Tip Think of V as an upside down A for39 All Think of El as a backwards E Ther39e Exists Propositions A atomic elementary proposition is the underyhg meaning of a simple declarative sentence which is either39 true or39 false The truth or39 falsehood of a proposition is indicated by assigning it one of the truth values T for39 true or39 F for39 false uwf apt1 Jamal5 mu 5 LII0f LII0f apt1 Jamal5 5 uwf upcy apt1 J2Dl 5 LII0f 5221 IJz pup 2ud 22nd5 lo 155L02 ppmDy 710 5152 U22J5J9I2 211 1502 21 54qu un5 211 p2p00q twpM 210 gnuwow suomsodoad 2dumgtlt3 Mom 401 LIA100 md C39Sb39qpunowns 54 13 0w 0 may 54 710 JIM07 214 a5ump 2101 34 310 21 up 1an 0 LIA01 01 1an 010 auo 710 do uo dn pmq Kat1 5102 710 S39LQlIUJ JaIO suomsodoad W 210 143ng saouawag Which are proposi rions Can pys fy Pys can fly Sparrows can fly Joe runs fasfer fmn Puffck Pa frck runs slower fmn Joe Pay your bills on me The dream ference of a circle Is equal 7 0 four mes ifs a fame fer Pr39oposiTionol connecTives These are The words Tho r we use To join o romic pr39oposi rions Toge rher39 To form compound pr39oposi rions EG In 1938 HITer seized Ausfra M in 1939 he seized former Czechoso vaka ana in 194 he affackea fIe former USSR m 57 I39 having a non aggre55on pacf wfh 397 Pr39oposiTionol connecTives Propositional logic has four connectives Nome Read as Symbol nega on 39nof conjunc an 39ana a fsjunc an 39or v IMPI39cafl39on 39I39f fhen 2 Interpretation of connectives Connective Interpretation negation p is true if and only if p is false A A conjunction pAq is true if and only if both p and q are true V A disjunction pvq is true if and only if p is true or39 q is true 3 An implication p gt q is false if and only if p is true and q is false Some more Terminology Expressions eiTher side of a conjuncTion are called conjuncTs pAq Expressions eiTher side of a disjuncTion are called disjuncTs pvq In The implicoTion p 2 q p is called The onTecedenT and q is The consequence Precedence of connecTives In complex pr39oposi rions br39ocke rs may be used To remove ombigui ry pA qv r versus p q v r By conven rion The order of precedence Br39ocke rs Nego rion Conjunction Disjunc rion Implico rion Formalisa rion S ra remen r In 1938 Hfer seized Ausfra and in 1939 he seized former Czechoso vaka in 194 he affackea fhe former USSR Mile 5f39 having a nonaggression pacf wfh if Formoliso rion con rinued A romic pr39oposi rions p In 1938 Hfer seized Ausfra q In 1939 Hfer seized former Czechoso vaka r39 In 194 Hfer affackea fne former USSR 5 In 194 Hfer had a nonaggression pacf mm fne former USSR For39moliso rion in Pr39oposi rionol Logic p A q A r39 A s Formalisa rion Afhough b0 1 Sfaney E Gordon are M young Sfaney has a be ffer chance of W g fhe nexf bowMg foumamen f desgfe Gordon s considerable experience Formoliso rion con nued A romic pr39oposi rions p 5 faney is young q 60rdon is young r39 5 faney has a be er chance of Winn g ve nexf bowMg four7amer s 60rdon has considerable experence in bowMg For39moliso rion in Pr39oposi rionol Logic p A q A r39 A s NegoTion and oTomic proposi rions No re Tho r for39 firs r o romic pr39oposi rion I chose Sfaney is young and no r Sfaney is 1707 young Summary of LecTure 1 Who r is for39mol logic Propositional logic Pr39oposi rions Pr39oposi rionol connectives For39moliso rion of arguments


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