Chapter 1 Axioms of the Real Number System

1 Chapter 1 Axioms of the Real Number Introductory Remarks: What constitutes a proof?One of the hurdles for a student encountering a rigorous calculus course for the first time,is what level of detail is expected in a proof. If every statement is completely justified, theproof that 1+1=2 takes over 200 pages of Principia Mathematics (Russell & Whitehead).So obviously, some things must be taken for granted. Just how much? Knowing the answerrequires some instructor and student to agree on what is to be assumedand what is to be , precisely what a proof is is stated explicitly. For the recordDefinition a finite sequence of statements, each of which is an axiom, oneof the hypotheses of the theorem, or follows from the preceding statements of the proof byelementary rules of inference, and the last statement of the proof is the conclusion of the end of this Chapter is an example of detailed mathematical reasoning, beyondwhat we shall require inthis text, but detailed enoughto indicate to thereader what truthsrest upon what 1.

2 Axioms OF THE REAL Number Propositional Logic and the Predicate Propositional LogicWe shall often need to prove sentences of the formp= q( )wherepandqare propositions . Apropositionis a statement, like 2 is an integer or 4 is a prime .Of course, the first of these propositions is true, and the second is meaning of Equation is taken as ifpis true, thenqis also true. We shallunderstand that this sentence is to be regarded as true ifpandqare both true, or ifpisfalse, regardless of thetruthvalueofq. This hasthesame logicalvalueas ifqisfalse, thenpalso must be false. (See Exercise 1.) That is, q= p,( )where qstands for notq, the proposition which is true whenqis false, and which is false whenqis (involvingpropositionsp,q,..)havethe samelogicalvalue (orareequivalent) is tosay that they aresimultaneously either bothtrueor bothfalse, regardlessof the truth value ofp,q.

3 The form in Equation is called thecontrapositiveform of the sentence p= q. Youshouldtakeafewminutestoconvince yourself thatthetwoformshavethesamelogicalvalue. In many places in the book, we shall prove p= q byproving q= p . Insummary, then,p= qif and only if q= represent pandq byp q, PROPOSITIONAL LOGIC AND THE PREDICATE CALCULUS3which will be true if and only if bothpandqare true. We represent porq byp q,which will be true if and only if (1)pis true or (2)qis true or (3) bothpandqare the exercises, you will be asked to verify that p= q is logically equivalent to theexpression p q. Exercise 1a. Make a truth table forp= q, by enumerating each of the four cases(true,true),(true,false), (false,true),and(false,false)ofvaluesfor (p,q).

4 Thendecideforeach ofthecaseswhetherp= qistrueorfalse, andenter itinthesecond columnin the q q= p p qTTTFFTFFb. Using the same truth table as in Part (a), decide for each of the four cases whether q= pistrueorfalse. Enteryourvaluesinthethirdcolumnofthetabl e. Comparewithp= that two expressions are equivalent if they have the same truthvalue for all possible truth values of the variables (in this casepandq.)c. Using the same truth table as in Part (a), decide for each of the four cases whether p qis true or false. Compare withp= form of proof we shall employ isproof by contradiction:If the assumption of the proposition pleads to a contradiction ( 0 = 1 ), then wemay conclude that pis false, thatpis 1 The familiar proof, due to Euclid, that there are infinitely many primes, pro-ceeds as follows:Suppose there were only finitely manyprimes, and we list ALL of them:p1,p2.

5 ,pk4 Chapter 1. Axioms OF THE REAL Number SYSTEMNow consider the integern=1+ claim thatnis also prime, because for anyi,1 i k,ifpidividesn, , it would divide their difference, 1, impossible. Hence theassumption thatpidividesnis false (since it led to contradiction), and hencepidoes notdividen, foranyi. Hencenisprime. But we listed ALL the primes above, asp1,..pk,andnisnot among them (since it is larger than eachof them). This contradicts the assumptionthat we listed all the primes, and hence our assumption that there were only finitely manyprimes is false. Hence there are infinitely many the above proof, note that we actually used the principle of proof by form of proof by contradiction written in propositional logic is the following:( p= FALSE) = qandq= pthenpandqare logically equivalent, and we writep qorp Predicate CalculusWe shall often make and prove statements of the form for everyx,P(x) which is written: xP(x) or there exists anxsuch thatP(x) which is written: xP(x).

6 Here we understand thatPis a proposition (see above) with a variablexin it, whichbecomes either true or false when we substitute a Number in forx. For exampleP(x)couldbe the statement12000 years later, Euler gave a totally different proof. Euler s proof is much more sophisticated, andwas the first proof which used Analysis to prove a result in Number theory, and thus introduced the field ofAnalytic Number Theory. See Chapter ** for his PROPOSITIONAL LOGIC AND THE PREDICATE CALCULUS5P(x) xis divisible by 2 ,a statement which is true when you substitute 4 forx,soP(4) is true, andP(122) as (3) is , for theP(x) we just defined, the statement xP(x)is false, while the statement xP(x)is understand the universe over which the quantifiers range to be understood fromthecontext.

7 For ourpurposes, typically theuniverse will bethe real numbers, or sometimesthe natural numbers. Other universes are , the statement xP(x)will be regarded astrueif there are NOx s in the universe, regardless of whatxis! Thus,we are going to regard the sentence All unicorns have 5 feet. as the statement xP(x)will be taken asfalseif there are nox s in the , the statement Some unicorns have 5 feet. will be taken as itisfalsethatforallx,P(x) istosaythattheremust besomexsuch thatP(x) does not hold. The negation of xP(x)6 Chapter 1. Axioms OF THE REAL Number SYSTEMis the sentence x P(x),and the negation of xP(x)is the sentence x P(x).Quantifiers are read left-to-right, like English, so we have the possibility of statementslike x yP(x,y).Supppose, for example, thatP(x,y) is the proposition x<y, where the variablesxandyare understood to range over the real numbers.

8 Then the statement x y(x<y)is true, since for every real numberxthere is another real Number ( +1)whichisgreater thanx. But the statement x y(x y)is false, since it asserts there is a Number (x) which is the smallest real Number . Notealso that as the quantifiers are read left-to-right, in x y(x<y),ydepends uponx, but in x y(x y),xdoes not depend on, we shall need tobefamiliar with reasoning such as, Above, wejust concludedthat in the domain of real numbers it is false that there is a Number (x) which is less thanevery real Number (y). Hence x y(x y),which is equivalent to x y (x y), , x y(x>y), PROPERTIES OFR, THE REAL NUMBERS:7which isthestatement thatforevery realnumber (x)thereisanotherrealnumberysmallerthan it ( 1.) Also when the existence of anxwith a certain property is asserted, we shall frequentlyname one, sayx0, as in the following.

9 F(x) is not continuous on all of [0,1] and hence there exists anx [0,1] such thatfis not continuous anx. Exercise 2 Construct the negation of p= q. Exercise 3 Construct the negation of x[P(x)= Q(x)]. Exercise 4 Construct the negation of x y[P(x,y)= Q(x,y)]. Exercise 5 Construct the negation of x y z[P(x,y,z)= Q(x,y,z)]. Exercise 6 The definition of fis continuous atx=a is the following: forevery >0thereexistsa >0suchthatforallx,if|x a|< then|f(x) f(a)|< . a. Writethisasalogicalexpression inthePredicate Calculus, identifying thecomponentpieces. (Assume that the universe of and is all positive numbers.)b. Construct the negation of fis continuous atx=a. Properties ofR, the Real The Axioms of a Field:The real numbersR=( , ) form a set which is also afield, as follows: There are twobinary operations onR, addition and multiplication, which satisfy a set of Axioms whichmake the setRacommutative group under addition: (all quantifiers in what followsare assumed to be over the universe of real numbers,R.)

10 1. For everyxandy,x+y=y+x.(Commutativity)8 Chapter 1. Axioms OF THE REAL Number SYSTEM2. For everyx,yandz,x+(y+z)=(x+y)+z.(Associativ ity)3. There exists an elementxsuch that for everyy,x+y=y. (We call this element theadditive identity, and after proving that it is unique, we label it 0. See Theorem a proof of uniqueness.)Restated: x[0+x=x]. (Identity)4. For everyxthere exists aysuch thatx+y=0.(Additive inverse). Note that fromthe order of the quantifiers,ydepends uponx. We usually denote thisyby x .The non-zero elements ofRform acommutative group under multiplication:1. For everyxandyx y=y x.(Commutativity)2. For everyx,yandz,x (y z)=(x y) z.(Associativity)3. There exists anxsuch that for everyy, x y=y. (We call this element the multi-plicative identity, andafterproving thatit isunique, we labelit 1.)