An Introduction to Advanced Mathematics

1 An Introduction to Advanced MathematicsM. YotovDecember 9, 2017 For .. Y ou Know Who!1 These Notes constitute a version of the courseMAA 3200 Introduction to Advanced Mathematicstaught by the author at the Department of Mathematics and Statistics of FIU. The concepts ofclasses, sets, relations, and functions are introduced and studied with rigour. Care is taken in moti-vating the Introduction of the Zermelo-Fraenkel axioms. The natural numbers and the principle of(finite) mathematical induction are discussed in detail. The integer, rational, and real numbers areconstructed and thoroughly discussed. As a brief Introduction to Advanced Calculus, the classicaltopology, convergent sequences of real numbers, and continuity of real valued function of a realvariable are studied. The Bolzano- weierstrass Theorem, Intermediate Value Theorem, and weier - strass s Theorem are send comments and corrections to the author at.

2 C 2016 Single paper copies for noncommercial personal use may be made withoutexplicit permission from the copyright Logic. Language of Propositions .. Propositional Expressions, Tautologies .. Propositional Functions and Quantifiers .. Methods of Proof ..142 Undefined Terms, First Axioms .. The Algebra of Sets ..283 Properties The Least Element Principle (LEP) forN.. The Principle of Math Induction .. Recursion, + and inN.. What Are Natural Numbers; Peano s Axioms .. Two Examples ..404 Relations, Functions, and Class Relations and Class Functions .. Relations from a Set to a Set .. Functions from a Set to a Set .. Pollency of Sets .. Equivalence Relations .. Orders ..715 Construction of the Standard Number The Integers .. The Rational Numbers .. The Real Numbers ..906 Topology on the Real The Classical Topology onR.

3 Sequences of Real Numbers .. Arithmetic Operations and a Relation on Sequences .. Intro to Cantor s Real Numbers .. 1127 The SetsF(X,Y), X,Y Limits of Functions .. Compact Subsets ofR.. Compact Subsets of a Topological Space .. Continuity of Functions .. 11938 Cantor s Real Preliminaries on Cauchy Sequences .. The Ordered FieldRC.. (RC, ) is Continuous .. 129 Index1324 Chapter 1 Logic. Language of PropositionsWe begin with an informal discussion ofpropositionsin Logic: the formal approach would leadus too far beyond the scope of the course. In the definition below, we are relying on the knowledgeof the reader about sentences in English. We are also assuming that the reader is able to tell if astatement, , a sentence containing a claim, is true or false. The latter depends on the context inwhich the statement is considered.

4 In this course, the context will be the one of propositionPis a statement, , a special sentence, which is either true is very important, for our considerations, that there are only two options for a statement:it caneither be true or false, and no third option is allowed. Many statements can be formulated(say, in English), but not every such is a proposition (according to the definition above)! To elevatethe status of a sentence from a statement to a proposition, the statement has to be, firstly, wellformulated and understood (all words and symbols in the used in the sentence have to be wellunderstood, and the criteria for telling if the statement if true or false have to be known), and,secondly, the statement has to be either true or false, no third option given. The latter is alsoexpressed by saying that the statement has to have a well determinedtruth value: T for true and F for false.

5 Example ) How are you doing? (This is not a statement)2) A lunar month is long less than 28 days. (This is a statement, but is not a proposition. Thereason is that, for instance, that there are several different lunar months, and there is no specificationof the one in the statement.)3) The one who is reading these notes now can read English.(This is a statement. It will be aproposition provided we know what can read English means.)4) There is only one even prime number. (This is a statement. For people who know what even andprime number is, it is a proposition. This proposition has truth value T.)2In our course, as already mentioned, we will be considering math propositions only. So, it is im-portant that the sentence, the statement, be formed by using mathematically well defined concepts,and that the claim in that statement be formulated in a mathematically appropriate way. To dothat formally, we need to havean alphabet(including math symbols), and alsorules how tocorrectly construct such sentences(so that they be propositions!

6 5 Constructing PropositionsWe explain now how to construct (more complex) propositions starting with given onesP,Q,..To do this we useconnectives , , , , , anddelimiters(,). Thus, one definesP Q, P Q, P, P Q, P Qbyassigning a truth value of each of the constructed propositions based on the truthvalues of the old propositions. This is conveniently done in truth tables. In words,P Q( PorQ ) is false only if bothPandQare false;P Q( PandQ ) is true only if bothPandQare true; P( notP ) is true only ifPis false;P Q( ifP,thenQ ) is false only ifPis true andQis false;P Q( Pif, and only if, Q ) is true only if bothPandQhave the same truth value:both are true or both are the statementsP= Peter needs help with Calculus 1 Q= Peter calls John and suppose they are propositions (that is, we know perfectly well what they mean, as well as howto logically evaluate them). To save some further explanations on who Peter and John are, whatCalculus 1 is, and what the rules for finding the truth values of P and Q are, assume P is true, andQ is false.

7 We have the following. P Qcan be expressed in English as Peter needs help with Calculus 1 or he calls John , oras Either Peter needs help with Calculus 1 or he calls John . This is a true statement. P Qcan be expressed in English as Peter needs help with Calculus 1 and he calls John .This is a false statement. Pcan be expressed in English as Peter does not need help with Calculus 1 . This is afalse statement. P Qcan be expressed in English as If Peter needs help with Calculus 1, (then) hecalls John , or as When/whenever/in case Peter needs help with Calculus 1, he calls John , or as Should Peter need help with Calculus 1, he calls John , or as Peter calls John only if he needshelp with Calculus 1 , or as .. This is a false statement. P Qcan be expressed in English as Peter calls John precisely when he needs helpwith Calculus 1 , or as Peter Calls John when he needs help with Calculus 1 and in no othercircumstances , or as the only reason for Peter to call John is if he needs help with Calculus 1 , oras.

8 This is a false newly defined propositions have names:P Qis thedisjunction ofPandQ,P Qis theconjunction ofPandQ, Pis thenegation/denial ofP,P Qis called aconditionalproposition withhypothesis/condition/antecedentthe propositionPand withconclusion/consequent- the Qis called using conditionals, one can express every theorem in Mathematics . In Logic, the conditionalP Qcan be read either as QifP (another way of saying ifP, thenQ ), or as Ponly ifQ In English, a conditional can be explained in many, equivalent, but looking quite unrelated, students have to learn how to detect a conditional in such English conditionalsQ Pand Q Pare calledconverseandcontrapositivetoP Q, biconditionalP Qis often read as, and replaced by the English expression, Pif, and onlyif,Q or even with the shorter PiffQ .6 The compound propositionP Qis calledinclusive or . This is, because its value is truenot only when P or Q is true, but also when both, P and Q, are true.

9 On the other hand, in ev-eryday speech, when people say something or something else , they usually understand the or exclusively: either something or something else , but not both. Turns out that in math similarto those real life situations appear, and the mathematicians have introduced a special notation forthe compound proposition which means that:exclusive or (XOR). Formally, given propositionsPandQ, denote byP XOR Qthe proposition (P Q) (Q P). The value of this propositionis true if, and only if, exactly one ofPandQis Propositional Expressions, TautologiesIn our definition of new propositions based on old ones, we didn t need to know what the old propo-sitions meant. That is, we didn t need to know what their truth values were. We just exhausted thecollection of all combination of values those propositions could have, and determined the value of thenew ones. What this formally means is that we simply considered the letters denoting the old propo-sitions asvariableswho take on values true or false.

10 This suggests the definition ofpropositionalexpressionsbased on analphabet(p, q, r,.., x, y, z,..,X1,X2,X3,..,Y1,Y2,Y3,..), wherewe take symbols for our variables-propositions from,connectives( , , , , ), anddelimiters((,)), and then using the five ways of constructing compound propositions from the previous sub-section. Note that the variables are just symbols. In what follows we will reserve the capital lettersfrom the English alphabet such as P, Q, R,X1,X2,.., to denote propositions known from thecontext, or variables whose values are propositions. We specifically designate two special symbols,the constants,TandF. to be variables taking only value true in the former case, and only falsein the latter the big difference between (compound) propositions and propositional expressions. Theformer always have a truth value - they are propositions after all! The latter arenotpropositionsin general!