Handout: Natural deduction for first order logic
|
|
- Jocelin Lamb
- 6 years ago
- Views:
Transcription
1 MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes re intended to replce Section 24 in Enderton (Enderton uses deduction system bsed on Hilbert-style xioms) 1 Well formed formuls We will dopt the definition of terms nd well-formed formuls from erlier following bbrevitions: Recll the ( ) ( ) ( ) x x 2 The inference rules ( quick reference) For reference we list ll the inference rules here In the next few sections we will explin the nottion nd the side conditions 21 Rules for the sententil connectives Impliction introduction nd elimintion I E Reductio d bsurdum RAA Assumption Rule 1
2 22 Rules for the quntifiers Inference rules for where for we require: y x y x E x t x is not free in ny open (uncovered) hypotheses bove this rule, y is substitutble for x in, nd either y is not free in, or y = x nd for E we require: t is substitutble for x in 23 Rules for equlity Inference rules for = t = t R t = s S s = t t = s s = r T t = r t = s f t = f s fsub t = s P s P t P Sub 3 Nturl deduction rules for the sententil connectives We dopted the following (fmilir) inference rules for the sententil connectives nd The only difference is now nd represent first-order formuls insted of sententil formuls Inference Rules for Reductio d bsurdum I Here is n bbrevition for Assumption Rule RAA E 2
3 4 Derived inference rules for sententil logic All the other inference rules for the sententil connectives cn be derived from these four rules Such rules, formed by combining other rules, re clled derived rules Exmple 1 The flse elimintion rule (ex flso), E is just specil cse of RAA where we do not cncel the ny ssumptions Exmple 2 The negtion introduction nd elimintion rules I E in their unbbrevited forms re I E These re just specil cse of the impliction introduction nd elimintion rules However not ll derived rules re specil cses of nother rule Exmple 3 The disjunction introduction nd elimintion rules, I 2 I 2 E re bbrevitions for the rules 3
4 These rules cn be derive from our originl four rules nd the derived rules s follows y x E I x I x I x I y E RAA x Notice tht the derivtion of hs the sme conclusion nd the only uncovered ssumption (hypothesis) is The derivtion of E lso hs the desired conclusion nd hypotheses Moreover, it uses tht we hve derivtions of from nd These derivtions don t need to go on the top of the proof Remrk 4 One wy to think bout these derived rules is tht if we hve proof tht uses, sy, E we cn replce it with proof without E by replcing every instnce of the rule E with its derivtion Exercise 5 Show the inference rules for nd re derivble Exercise 6 Show tht we cn lso derive the following inference rules which we will cll double negtion introduction, double negtion elimintion nd lw of the excluded middle I E LEM 5 Substitution In order to give the deduction rules for quntifiers, we will need to define two concepts: 1 α x t, mening the formul one gets by substituting every free instnce of the vrible x with the term t 2 t is substitutble for x in α, mening tht it is OK to substitute t for x While first concept is quite nturl, it is helpful to give recursive definition Definition 7 Given wff α, vrible x, nd term t, define α x t recursively s follows If α is the tomic formul P u 1 u n, then α x t is P (u 1 ) x t (u n ) x t where (u i ) x t is the term u i with ll x replced by t (I will leve defining (u i ) x t recursively s homework exercise) 4
5 ( ) x t is ( x t x t ) ( x ) x t is x (Notice, x is not free in x ) ( y ) x t is y x t, where y is vrible different from x Exercise 8 Show tht the following hold for our bbrevited formuls: ( ) x t is ( x t ) ( ) x t is ( x t x t ) where is ny of,, ( x ) x t is x ( y ) x t is y x t, where y is vrible different from x Exmple 9 1 x x = 2 (Qx x P x) x y is (Qy x P x) (We only replce the free occurrences of x) 3 (Qx P x) x fxy is (Qfxy P fxy) (Notice, x is still free in this formul We replced the two free x in the originl formul, then we stop) 4 ( y (x y)) x z is y (z y) 5 ( y (x y)) x y is y (y y) (Notice the first is true in ny structure of crdinlity 2, but the second is not true in ny structure!) We would like to sy if α is true, then so is α x t However, we need dditionl conditions to void the pthology of (5) Informlly, we sy t is substitutble for x in α if there is no vribles in t which re cptured by quntifier fter the substitution Definition 10 Define t is substitutble for x in α recursively s follows 1 For α tomic, t is lwys substitutble for x in α t is substitutble for x in ( ) iff t is substitutble for x in both nd t is substitutble for x in y iff either x is not free vrible of y (this includes the cse where y is x), or y is not vrible in t, nd t is substitutble for x in Exercise 11 Prove x is lwys substitutble for x in α (Use induction on wffs) Exercise 12 Show tht the following hold for our bbrevited formuls: t is substitutble for x in ( ) iff t is substitutble for x in 1 Mny books use the phrse t is free for x in α 5
6 t is substitutble for x in ( ) iff t is substitutble for x in both nd, where where is ny of,, t is substitutble for x in y iff either x does not occur free in y (this includes the cse where y is x) y does not occur free in t, nd t is substitutble for x in Remrk 13 We my still use α x t even when x is not substitutble for x in t 6 Rules for quntifiers 61 The rules for the universl quntifier Introduction nd elimintion rules for the universl quntifier where for we require: y x y x E x t x is not free in ny open (uncovered) hypotheses bove this rule, y is substitutble for x in, nd either y is not free in, or y = x nd for E we require: t is substitutble for x in We will crefully discuss these side conditions, but first some motivtion 62 Motivtion 621 Similr to nturl lnguge proofs These rules re designed to mimic informl nturl lnguge proofs For the introduction rule, ssume I wnt to prove x In n informl nturl lnguge proof, I would strt by fixing n rbitrry object x nd then giving proof tht holds of x (where ssumedly x is free vrible in ) Finlly, since x is rbitrry (I mde no specil ssumptions bout wht x is or wht properties it hs) then I cn deduce x This is wht the rule does The side conditions re there to ssure tht x relly is rbitrry For the elimintion rule, gin think of informl nturl lnguge proof If I know hold for ll x, then I my ssume it holds for some specific object (which in this cse is given by the term t) The side condition is there to void technicl issues 6
7 622 Similr to the conjunction rules Recll tht behves like n infinite conjunction Now compre our rules for with our rules for : I E 1 The reder would do well to crefully meditte on the similrity (nd differences) between the rules for nd those for 63 The side conditions The rules for come with subtle, but very importnt side conditions We will discuss them one t time 631 Side conditions for : x is not free in ny open (uncovered) hypotheses bove this rule To relly understnd wht this mens nd why it is importnt let us give n exmple of good proof nd bd proof This is good proof: P x P y I P x P y P x P x P y x (P x P x P y) There is only one uncovered hypothesis nd it is does not contin x s free vrible This is BAD PROOF : x 0 x (x 0) We re ssuming x 0, nd then somehow getting the conclusion tht no number is equl to 0 Clerly flse if 0 is constnt in our lnguge This hppened, becuse we pplied when x 0 ws n open hypothesis contining the free vrible x This is lso BAD PROOF : x 0 x (x 0) x 0 x (x 0) Even though x 0 is now covered, tht is becuse it mtches with something below our ppliction of The subproof, where we pply t the bottom, is the sme s (1) nd hs x 0 s open either y is not free in, or y = x E I E 2 (1) 7
8 When we use we cn use the sme vrible x for the quntifier tht we re generlizing (this is the y = x cse), eg P x E P x P x x (P x P x) We my lso use new vrible y s long s it is free in P x P x P x y (P y P y) E This my even pply generlize on vrible not used in For exmple, P x P x P x y (P x P x) here I m generlizing on y even those it does not exist in P x P x However, I cn cnnot use vrible tht is free in This is BAD PROOF : E P x E P x P x y 0 I (P x P x) y 0 y ((P y P y) y 0) See tht we ve gin concluded tht no number is 0?! y is substitutble for x in As we hve lredy seen things cn go wrong when we try to substitute one vrible for nother if tht vrible lredy conflicts with bound vrible Exercise 14 Come up with n exmple showing why we need this rule Find proof which follows ll the rules except this one nd derives something invlid 632 Side conditions for E: t is substitutble for x in Given x we cn replce x with ny term t tht is substitutble for x in For exmple, this is good: x y (y x) E y (y 1 + 1) However this is BAD PROOF : x y (y x) E y (y y) The term y is not substitutble for x in y (y x) 8
9 64 Derived rules for the existentil quntifier The derived rules for re s follows Introduction nd elimintion rules for the existentil quntifier y x x t x I y E where for I we require: t is substitutble for x in nd for E we require: x is not free in nor in ny open hypotheses in the proof x is substitutble for y in, nd either x is not free in, or y = x y x other thn y x, Agin these rules cn be motivted by nturl lnguge proofs If I know there exists solution to y 5 y + 1 = 0, then I cn use this number to prove other things I do this by giving this number nme, sy x Then I prove things bout x only using the ssumption tht x 5 x + 1 = 0 For exmple, I cn prove there exists z such tht 2z 5 + 2z 2 = 0 This is the ide behind E Also, notice the similrity between these rules nd the rules for Here we derive the rules for the existentil quntifier y x b x E x t x t E I x y y x E I b y E RAA Exercise 15 Explore wht would go wrong if we violte ny of the side conditions in I or E 9
10 7 Rules for equlity The rules for equlity re s follows Inference rules for = t = t R t = s S s = t t = s s = r T t = r t = s f t = f s fsub t = s P s P t P Sub In these three rules t, s, nd r re terms in our lnguge The first three rules sy tht = is n equivlence reltion The lst two rules sy tht functions f nd predictes P respect equlity The nottion t nd s is shorthnd The longer version of the rules re t 1 = s 1 t n = s n fsub ft 1 t n = fs 1 s n t 1 = s 1 t n = s n P t 1 t n P Sub P s 1 s n where f nd P re n-plce In prticulr, for constnts c we hve specil cse of the fsub rule, csub c = c which is redundnt with the reflection rule R Exmple 16 Assume we hve constnts 0, 1, unry function, nd binry functions +, This next proof sys tht if x = 1 nd y = 0 then ( x + 0) + (y + z) = ( 1 + 0) + (0 + z) R x = 1 y = 0 z = z Sub R +Sub x = 1 0 = 0 y + z = 0 + z +Sub Sub x + 0 = (y + z) = (0 + z) +Sub ( x + 0) + (y + z) = ( 1 + 0) + (0 + z) Exmple 17 Assume we hve the predicte P nd the constnt c This next proof sys tht if x = c nd x Qx (P x ) holds, then so does x P x (P c ) (I replced ll free x with c) x = c x Qx (P x ) x Qx S b E c = x P c P Sub P x P x E I b P c I x Qx (P c ) To simply the tedium of the lst two exmples, consider these two derived rules Exercise 18 Let x = x 1,, x n denote list of unique vribles Let s = s 1,, s n be list of terms Use t x s to denote the term t with ll the vribles x 1,, x n replced respectively by the terms s 1,, s n Use x t to denote the formul with x 1,, x n replced by t 1,, t n Define t x s nd x t recursively, nd then derive these two rules: s = r t x s = t x r Sub t = s x s x t Sub 10
Introduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationMath Lecture 23
Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in
More informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationCHAPTER 4 MULTIPLE INTEGRALS
CHAPTE 4 MULTIPLE INTEGAL The objects of this chpter re five-fold. They re: (1 Discuss when sclr-vlued functions f cn be integrted over closed rectngulr boxes in n ; simply put, f is integrble over iff
More informationLecture 3: Equivalence Relations
Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts
More informationLinear Algebra 1A - solutions of ex.4
Liner Algebr A - solutions of ex.4 For ech of the following, nd the inverse mtrix (mtritz hofkhit if it exists - ( 6 6 A, B (, C 3, D, 4 4 ( E i, F (inverse over C for F. i Also, pick n invertible mtrix
More informationUniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More information1 Sets Functions and Relations Mathematical Induction Equivalence of Sets and Countability The Real Numbers...
Contents 1 Sets 1 1.1 Functions nd Reltions....................... 3 1.2 Mthemticl Induction....................... 5 1.3 Equivlence of Sets nd Countbility................ 6 1.4 The Rel Numbers..........................
More informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
More informationStrong Bisimulation. Overview. References. Actions Labeled transition system Transition semantics Simulation Bisimulation
Strong Bisimultion Overview Actions Lbeled trnsition system Trnsition semntics Simultion Bisimultion References Robin Milner, Communiction nd Concurrency Robin Milner, Communicting nd Mobil Systems 32
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationCS 188 Introduction to Artificial Intelligence Fall 2018 Note 7
CS 188 Introduction to Artificil Intelligence Fll 2018 Note 7 These lecture notes re hevily bsed on notes originlly written by Nikhil Shrm. Decision Networks In the third note, we lerned bout gme trees
More informationMath Solutions to homework 1
Mth 75 - Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
More informationCMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014
CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA
More informationHeat flux and total heat
Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationMAT 215: Analysis in a single variable Course notes, Fall Michael Damron
MAT 215: Anlysis in single vrible Course notes, Fll 2012 Michel Dmron Compiled from lectures nd exercises designed with Mrk McConnell following Principles of Mthemticl Anlysis, Rudin Princeton University
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More information1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.
Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationCS 275 Automata and Formal Language Theory
CS 275 Automt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Problem (II) Chpter II.5.: Properties of Context Free Grmmrs (14) Anton Setzer (Bsed on book drft by J. V. Tucker nd K. Stephenson)
More informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More information(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer
Divisibility In this note we introduce the notion of divisibility for two integers nd b then we discuss the division lgorithm. First we give forml definition nd note some properties of the division opertion.
More informationHomework Solution - Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationLecture 2: Fields, Formally
Mth 08 Lecture 2: Fields, Formlly Professor: Pdric Brtlett Week UCSB 203 In our first lecture, we studied R, the rel numbers. In prticulr, we exmined how the rel numbers intercted with the opertions of
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationLine Integrals. Chapter Definition
hpter 2 Line Integrls 2.1 Definition When we re integrting function of one vrible, we integrte long n intervl on one of the xes. We now generlize this ide by integrting long ny curve in the xy-plne. It
More informationCS 275 Automata and Formal Language Theory
CS 275 Automt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Problem (II) Chpter II.6.: Push Down Automt Remrk: This mteril is no longer tught nd not directly exm relevnt Anton Setzer (Bsed
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More information1 Nondeterministic Finite Automata
1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
More informationThe Henstock-Kurzweil integral
fculteit Wiskunde en Ntuurwetenschppen The Henstock-Kurzweil integrl Bchelorthesis Mthemtics June 2014 Student: E. vn Dijk First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. A. vn der Schft
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationLecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014
Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t Urbn-Chmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationAnalytical Methods Exam: Preparatory Exercises
Anlyticl Methods Exm: Preprtory Exercises Question. Wht does it men tht (X, F, µ) is mesure spce? Show tht µ is monotone, tht is: if E F re mesurble sets then µ(e) µ(f). Question. Discuss if ech of the
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationHomework 11. Andrew Ma November 30, sin x (1+x) (1+x)
Homewor Andrew M November 3, 4 Problem 9 Clim: Pf: + + d = d = sin b +b + sin (+) d sin (+) d using integrtion by prts. By pplying + d = lim b sin b +b + sin (+) d. Since limits to both sides, lim b sin
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More informationReview on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.
Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5. - 5.3) Remrks on the course. Slide Review: Sec. 5.-5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description
More informationLinearly Similar Polynomials
Linerly Similr Polynomils rthur Holshouser 3600 Bullrd St. Chrlotte, NC, US Hrold Reiter Deprtment of Mthemticl Sciences University of North Crolin Chrlotte, Chrlotte, NC 28223, US hbreiter@uncc.edu stndrd
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More information5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship
5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is
More informationCSCI FOUNDATIONS OF COMPUTER SCIENCE
1 CSCI- 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 My 7, 2015 2 Announcements Homework 9 is due now. Some finl exm review problems will be posted on the web site tody. These re prcqce problems not
More information