University of New EnglandSchool of Science and TechnologyMATH101ALGEBRA ANDDIFFERENTIAL CALCULUSLecture Notes Part 1Trimester 1, 2015c University of New EnglandCRICOS Provider No: 00003G

CONTENTSiContentsPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iiiLecture 1.1 Mathematical Language and Proof . . . . . . . . . . . . . . .1Lecture 1.2 Important Types of Theorems and Proof . . . . . . . . . . .7Lecture 1.3 Sets and Functions . . . . . . . . . . . . . . . . . . . . . . . . 14Lecture 1.4 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Lecture 1.5 Some Properties of Real Numbers . . . . . . . . . . . . . . . 27Lecture 1.6 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 33Lecture 1.7 Complex Numbers (continued) . . . . . . . . . . . . . . . . . 40Lecture 1.8 Functions on R . . . . . . . . . . . . . . . . . . . . . . . . . . 48Lecture 1.9 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Lecture 1.10 Limits and Continuous Functions . . . . . . . . . . . . . . . 63Lecture 1.11 Continuous Functions . . . . . . . . . . . . . . . . . . . . . 69Lecture 1.12 More on Continuity . . . . . . . . . . . . . . . . . . . . . . . 76Lecture 1.13 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Lecture 1.14 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . 88Lecture 1.15 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


iiiPrefaceMathematics today is a vast enterprise. Advances and breakthroughs have beenpainstakingly built on the structure(s) erected by earlier mathematicians. The history of mathematics is quite different from the that of other human endeavours. Inother fields, previously held views are typically extended or proved wrong with eachadvance there is a process of correction and extension. “Only in mathematics isthere no significant correction – only extension”.The work of Euclid has certainly been extended many times. Euclid, however,has not been corrected – his theorems are valid today and for all time! The otherremarkable thing about mathematics is its extraordinary utility in describing andquantifying the world around us. Mathematics is the language of the sciences, bothnatural and social. This forces mathematics to be abstract, since it must embracetheories from physics, economics, chemistry, psychology, etc. Mathematics is sowidely applicable precisely because of — not despite — its intrinsic abstractness.MATH101 is the first half of the MATH101/102 sequence, which lays the foundation for all further study and application of mathematics and statistics, presentingan introduction to differential calculus, integral calculus, algebra, differential equations and statistics, providing sound mathematical foundations for further studiesnot only in mathematics and statistics, but also in the natural and social sciences.Achieving this, requires a brief, preliminary foray into the basics of mathematics,because much of the material requires a high degree of abstract reasoning, ratherthan rote learning of computational techniques.A rigorous approach to the basics provides a deeper understanding of the wholestructure. The assumptions upon which the structure is built are thereby clarifed,with both the scope and limitations of the intellectual framework made readilyunderstandable. Moreover, this deeper understanding, does not come at the expenseof applicability. Quite the contrary!One consequence of providing sound fundamentals is that there is considerabletime devoted to matters whose importance and applicability is not immediately obvious. But such study of these fundamental areas of mathematics is also stimulating.If you enjoy puzzles here is an “intellectual game” par excellence. A game playedwithin a rigid framework of rules, but with unlimited scope for creativity in thesearch for problems and the solutions to problems.This is the first of three parts of the lecture notes which together constitute theunit material for MATH101. These notes were originally prepared by Chris Radfordand have been revised by Shusen Yan and others.

ivPrefaceThe reader is invited and encouraged to point out any mistakes (s)he finds. Ihope you enjoy the challenges the unit offers and that you experience a sense ofachievement at the end.Bea BleileUNE

Lecture 1.1 Mathematical Language and ProofLecture 1.11Mathematical Language and ProofIntroductionWhat is research in mathematics? A mathematician would answer “provingtheorems”. The language and etiquette of mathematics has evolved over a longperiod of time. Terms such as theorem, axiom, definition and proof have a universaland well understood meaning. It is these conventions and definitions we want toexamine in this lecture.Basic TerminologyStatement: A statement is a sentence which is either true or false(according to some previously accepted criteria). Statements do notinclude exclamations, questions or orders. A statement cannot betrue and false at the same time.A statement is simple when it cannot be broken down into other statements. Astatement is compound when it contains more than one simple statement. Example “I will have a BBQ and it will rain” is a compound statement consistingof two simple statements. “I will have a BBQ” and “It will rain”.Definition: A definition is a statement of the precise meaning of aword, phrase, mathematical symbol or concept.In trying to understand a piece of mathematics it is important to have a goodworking understanding of the initial definitions. You need to look at examples whichsatisfy and do not satisfy the definitions.Theorem: A theorem is a mathematical statement that can be provedtrue by a chain of logical argument based on assumptions that aregiven or implied in the statement of the theorem.A theorem will give a deeper insight into the structure of a piece of mathematics.Lemma: A lemma is a preliminary theorem read in the proof ofanother theorem.

2Lecture 1.1 Mathematical Language and ProofSome theorems can have proofs that are long and intricate. It is useful in suchcases to break the proof into intermediate steps which are separated out as lemmaswhich lead into the main result.Corollary: A corollary is a theorem that is a natural consequenceof a preceding theorem. Generally, a corollary will follow in a relatively easy and straight-forward way from the previous theorem orproposition.Proof: A proof of a proposition, theorem, lemma or corollary is asequence of logical reasoning. It is based on the given assumptionsor hypothesis and aims to establish the truth of the proposition ortheorem.Basic Techniques of ProofImplicationConsider the following compound statement: “If I pass MATH101 then I will doMATH 102”. Under what circumstances is this statement true or false?Let’s have a look at the two simple statements making up this compound statement.A. I pass MATH101.B. I will do MATH102.For statement A there are two possibilities,1. I do pass MATH101 (A is true).2. I fail MATH101 (A is false).There are also two possibilities for B,1. I will enrol in MATH102 (B is true).2. I will not enrol in MATH102 (B is false).

Lecture 1.1 Mathematical Language and Proof3So we have to consider four eTrueFalseAs well as the truth or otherwise of A and B, individually, we must consider thetruth of the original compound statement which takes the form of an implication.This compound statement is “If A then B”. We enlarge our table above to include acolumn for “If A then B”. We must decide for each of the four cases if the compoundentry is True or False.Case 1: I pass MATH101 and enrol in MATH 102. The implication is True.Case 2: I pass 101 and do not enrol in 102. Implication is False.Case 3: I do not pass 101 but still enrol in 102. My original statement is not alie, it is not a falsehood. Implication is True.Case 4: I do not pass 101 and do not enrol in 102. Again, my original statementis not a lie or a falsehood. Implication is eIf A then BTrueFalseTrueTrueThere is only one case when the implication is false, ie. when A is true and B isfalse.The implication “If A then B” is true if we can prove that it isimpossible to have A true and B false at the same time.This means if we assume “A implies B” to be true and we also assume that A istrue then we must conclude that B is true.The statement “if A then B” is equivalent to the statement “A is asufficient condition for B” and to the statement “B is a necessarycondition for A”.

4Lecture 1.1 Mathematical Language and ProofNotice that in a statement of the form “A implies B” the hypothesis, part A, isclearly distinguished from the conclusion, part B. A direct proof of a mathematicalimplication, “A implies B”, proceeds on the assumption that the hypothesis A istrue. The proof will work via a series of logically connected steps to obtain theconclusion B. Example Prove that the sum of two prime numbers larger than 2 is an evennumber.Solution We formulate this as an implication:“If p and q are prime numbers larger than 2 then p q is even”.The hypothesis “p and q are prime numbers larger than 2” assumes that we knowwhat prime numbers are. A prime number is a natural or counting number divisibleonly by itself and one – 2, 3, 5, 7, 11, 13, 17, 19, 23 are prime numbers.So we are to assume p and q are prime numbers bigger than 2. We must thenconstruct a series of arguments which lead directly to the conclusion that p q isdivisible by 2 (i.e. p q is even).Firstly, we see that p and q must be odd. We must be careful about this pointbecause 2 is prime but even! However, for primes greater than 2 we note that aprime cannot be divisible by 2; so it must have remainder 1 after division by 2. Sothere must be counting numbers n and m such thatp 2n 1q 2m 1This really just says that p and q are odd. We can now move to our conclusion,p q (2n 1) (2m 1) 2n 2m 2 2(n m 1).We conclude that p q is divisible by 2 and so p q is even.2This is a direct proof as we moved directly from the (assumed true) hypothesis,“p and q are prime numbers larger than 2” directly to the conclusion “p q is aneven number”.

Lecture 1.1 Mathematical Language and Proof5Proof by ContradictionRecall that the implication “If A then B” can only be false if we have A trueand B false. So if we were to assume B to be false and prove, as a consequence,that A is necessarily false, then we have proved the implication is true. We wouldhave shown that the only possible case with B false is A false which contradicts theassumed truth of the hypothesis. This is proof by contradiction.In general one tries to avoid using a proof by contradiction. If a direct proof isstraightforward then this is to be preferred – a direct proof usually provides moreinsight into the mathematical structure at hand. Example Prove that if p is a prime number larger than 2 then p 1 is not prime.Solution We formulate this statement as an implication with hypothesis “p is aprime number larger than 2” and conclusion “p 1 is not prime”.We will use a proof by contradiction. Assume the conclusion is false. We needthe negative of the statement “p 1 is not prime”. This means we must assumethat “p 1 is prime”.So our starting assumptions are now“p is a prime number larger than 2” (hypothesis)“p 1 is prime” (negative of the conclusion).We are now free to use any means at our disposal to find a contradiction based onthese two assumptions. If we can find such a contradiction then we have establishedthe truth of the original implication.We will use the result established in the previous example.We have two primes p and p 1 both larger than 2, so we know that p (p 1)must be even (this is the result of our earlier exercise). But this is clearly false asp (p 1) 2p 1 is an odd number.We have a contradiction. Our proof is complete, our assumption that the conclusion is false cannot hold, the conclusion must be true.2

6Lecture 1.1 Mathematical Language and Proof Exercises 11. Construct a direct proof for the last example, above.2. Use a direct proof technique to prove that if (a b)2 a2 b2 for all realnumbers b, then a 0.*3. Let n be a counting number. If 2n 1 is a prime number prove that n is alsoprime. Use a proof by contradiction to establish the truth of this statement.

Lecture 1.2 Important Types of Theorems and ProofLecture 1.27Important Types of Theorems and ProofIn your mathematical reading you will find that there are certain types of theoremwhose statement and proof follow a standard pattern. Our aim here is to look at acouple of the more important examples.Equivalence or “If and Only If” Theorems.The statement of theorems of this type takes one of the following possible forms“A is true if and only if B is true”“A is false if and only if B is false”“A is equivalent to B”.Notice that these forms are all basically equivalent – the truth (or falsity) of Aautomatically implies the truth (or falsity) of B and vice versa. So to prove such atheorem we have to produce two proofs:one proof for “A implies B” andone proof for “B implies A”.The first proof says A is sufficient for B (or B is necessary for A). The secondproof says B is sufficient for A (or A is necessary for B). In fact another common statement of equivalence is a theorem which takes the following form, “A is anecessary and sufficient condition for B”. Example Let n be a counting number. Then n is odd if and only if n2 is odd.Solution Let A be the statement “n is odd” and let B be the statement “n2 isodd”.We have to prove two implications1. A implies B2. B implies AFirstly, let’s examine A implies B. We’ll provide a direct proof. So we assumeA, that is, we assume n is an odd counting number. This means there is a countingnumber r such thatn 2r 1

8Lecture 1.2 Important Types of Theorems and Proof(any odd numb