2014 - University of Birmingham - Proofs and Math Reasoning

7/25/2019 2014 - University of Birmingham - Proofs and Math Reasoning

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Proofs and Mathematical Reasoning

University of Birmingham

Author:

Agata Stefanowicz

Supervisors:

Joe Kyle

Michael Grove

September 2014

cUniversity of Birmingham 2014
http://creativecommons.org/licenses/by-nc-nd/4.0/


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Contents

1 Introduction 6

2 Mathematical language and symbols 6

2.1 Mathematics is a language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Greek alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Words in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 What is a proof? 9

3.1 Writer versus reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Methods of proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Implications and if and only if statements . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Direct proof 11

4.1 Description of method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Hard parts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Fallacious proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Proof by cases 17

5.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 Hard parts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.3 Examples of proof by cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Mathematical Induction 19

6.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Versions of induction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3 Hard parts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.4 Examples of mathematical induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 Contradiction 26

7.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 Hard parts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.3 Examples of proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

8 Contrapositive 29

8.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.2 Hard parts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9 Tips 31

9.1 What common mistakes do students make when trying to present the proofs? . . . . . 31

9.2 What are the reasons for mistakes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.3 Advice to students for writing good proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 329.4 Friendly reminder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32



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10 Sets 34

10.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.2 Subsets and power sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3 Cardinality and equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.4 Common sets of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.5 How to describe a set? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.6 More on cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.7 Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.8 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

11 Functions 41

11.1 Image and preimage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.2 Composition of the functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.3 Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.4 Injectivity, surjectivity, bijectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.5 Inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.6 Even and odd functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

12 Appendix 47



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Foreword

Talk to any group of lecturers about how their students handle proof and reasoning whenpresenting mathematics and you will soon hear a long list of improvements they would wish for.And yet, if no one has ever explained clearly, in simple but rigorous terms, what is expected it ishardly a surprise that this is a regular comment. The project that Agata Stefanowicz worked on

at the University of Birmingham over the summer of 2014 had as its aim, clarifying and codifyingviews of staff on these matters and then using these as the basis of an introduction to the basicmethods of proof and reasoning in a single document that might help new (and indeed continuing)students to gain a deeper understanding of how we write good proofs and present clear and logicalmathematics. Through a judicious selection of examples and techniques, students are presentedwith instructive examples and straightforward advice on how to improve the way they produceand present good mathematics. An added feature that further enhances the written text is theuse of linked videos files that offer the reader the experience of live mathematics developed byan expert. And Chapter 9, that looks at common mistakes that are made when students presentproofs, should be compulsory reading for every student of mathematics. We are confident that,regardless of ability, all students will find something to improve their study of mathematics withinthe pages that follow. But this will be doubly true if they engage with the problems by tryingthem as they go through this guide.

Michael Grove & Joe KyleSeptember 2014



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Acknowledgements

I would like to say a big thank you to the Mathematics Support Centre team for the opportunityto work on an interesting project and for the help and advice from the very first day. Specialgratitude goes to Dr Joe Kyle for his detailed comments on my work and tips on creating thedocument. Thank you also to Michael Grove for his cheerful supervision, fruitful brainstorming

conversations and many ideas on improving the document. I cannot forget to mention Dr SimonGoodwin and Dr Corneliu Hoffman; thank you for your time and friendly advice. The documentwould not be the same without help from the lecturers at the University of Birmingham who tookpart in my survey - thank you all.

Finally, thank you to my fellow interns, Heather Collis, Allan Cunningham, Mano Sivanthara-jah and Rory Whelan for making the internship an excellent experience.



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1 Introduction

From the first day at university you will hear mention of writing Mathematics in a good style andusing proper English. You will probably start wondering what is the whole deal with words, whenyou just wanted to work with numbers. If, on top of this scary welcome talk, you get a number of

definitions and theorems thrown at you in your first week, where most of them include strange notionsthat you cannot completely make sense of - do not worry! It is important to notice how big differencethere is between mathematics at school and at the university. Before the start of the course, many ofus visualise really hard differential equations, long calculations and x-long digit numbers. Most of uswill be struck seeing theorems like a0 = 0. Now, while it is obviousto everybody, mathematiciansare the ones who will not take things for granted and would like to see the proof.

This booklet is intended to give the gist of mathematics at university, present the language used andthe methods of proofs. A number of examples will be given, which should be a good resource for furtherstudy and an extra exercise in constructing your own arguments. We will start with introducing themathematical language and symbols before moving onto the serious matter of writing the mathematicalproofs. Each theorem is followed by the notes, which are the thoughts on the topic, intended to givea deeper idea of the statement. You will find that some proofs are missing the steps and the purple

notes will hopefully guide you to complete the proof yourself. If stuck, you can watch the videos whichshould explain the argument step by step. Most of the theorems presented, some easier and othersmore complicated, are discussed in first year of the mathematics course. The last two chapters givethe basics of sets and functions as well as present plenty of examples for the readers practice.

2 Mathematical language and symbols

2.1 Mathematics is a language

Mathematics at school gives us good basics; in a country where mathematical language is spoken,after GCSEs and A-Levels we would be able to introduce ourselves, buy a train ticket or order a pizza.

To have a fluent conversation, however, a lot of work still needs to be done.

Mathematics at university is going to surprise you. First, you will need to learn the language tobe able to communicate clearly with others. This section will provide the grammar notes, i.e. thecommonly used symbols and notation, so that you can start writing your mathematical statements ina good style. And like with any other foreign language, practice makes perfect, so take advantageof any extra exercises, which over time will make you fluent in a mathematical world.

2.2 Greek alphabet

Greek alphabet - upper and lower cases and the names of the letters.

2.3 Symbols

Writing proofs is much more efficient if you get used to the simple symbols that save us writing longsentences (very useful during fast paced lectures!). Below you will find the basic list, with the symbolson the left and their meaning on the right hand side, which should be a good start to exploring furthermathematics. Note that these are useful shorthands when you need to note the ideas down quickly. Ingeneral though, when writing your own proofs, your lecturers will advise you to use words instead ofthe fancy notation - especially at the beginning until you are totally comfortable with the statementsif. . . , then. . . . When reading mathematical books you will notice that the word implies appearsmore often than the symbol = .



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A alphaB beta gamma deltaE epsilon

Z zetaH eta thetaI iota

K kappa lambdaM mu

N nu xiO omicron piP rho

sigmaT tauY upsilon phiX chi psi omega

Table 1: Greek letters

Quantifiers

(universal quantifier)

(existential quantifier)for all

there exists

Symbols in set theory

, or

unionintersection

subsetproper subset

composition of functions

Common symbols used when writing proofs and definitions

=

:=

: or|

or or

impliesif and only ifis defined as

is equivalent tosuch thattherefore

contradictionend of proof

2.4 Words in mathematics

Many symbols presented above are useful tools in writing mathematical statements but nothing

more than a convenient shorthand. You must always remember that a good proof should also includewords. As mentioned at the beginning of the paper, correct English (or any other language in whichyou are literate) is as important as the symbols and numbers when writing mathematics. Since it isimportant to present proofs clearly, it is good to add the explanation of what is happening at eachstep using full sentences. The whole page with just numbers and symbols, without a single word, willnearly always be an example of a bad proof!

Tea or coffee? Mathematical language, though using mentioned earlier correct English, differsslightly from our everyday communication. The classic example is a joke about a mathematician,



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3 What is a proof?

The search for a mathematical proof is the search for aknowledge which is more absolute than the knowledge accu-mulated by any other discipline.

Simon Singh

A proof is a sequence of logical statements, one implying another, which gives an explanation of whya given statement is true. Previously established theorems may be used to deduce the new ones; onemay also refer to axioms, which are the starting points, rules accepted by everyone. Mathematicalproof is absolute, which means that once a theorem is proved, it is proved for ever. Until proventhough, the statement is never accepted as a true one.

Writing proofs is the essence of mathematics studies. You will notice very quickly that from dayone at university, lecturers will be very thorough with their explanations. Every word will be defined,notations clearly presented and each theorem proved. We learn how to construct logical arguments andwhat a good proof looks like. It is not easy though and requires practice, therefore it is always temptingfor students to learn theorems and apply them, leaving proofs behind. This is a really bad habit (and

does not pay off during final examinations!); instead, go through the proofs given in lectures andtextbooks, understand them and ask for help whenever you are stuck. There are a number of methodswhich can be used to prove statements, some of which will be presented in the next sections. Hardand tiring at the beginning, constructing proofs gives a lot of satisfaction when the end is reachedsuccessfully.

3.1 Writer versus reader

Kevin Houston in his book[2] gives an idea to think of a proof like a small battle between thereader and the writer. At the beginning of mathematics studies you will often be the reader, learningthe proofs given by your lecturers or found in textbooks. You should then take the active attitude,which means working through the given proof with pen and paper. Reading proofs is not easy and mayget boring if you just try to read it like a novel, comfortable on your sofa with the half-concentrationlevel. Probably the most important part is to question everything, what the writer is telling you.Treat it as the argument between yourself and the author of the proof and ask them why? at eachstep of their reasoning.

When it comes to writing your own proof, the final version should be clear and have no gaps inunderstanding. Here, a good idea is to think about someone else as the person who would questioneach of the steps you present. The argument should flow and have enough explanations, so that thereader will find the answer to every why? they might ask.

3.2 Methods of proofs

There are many techniques that can be used to prove the statements. It is often not obvious at

the beginning which one to use, although with a bit of practice, we may be able to give an educatedguess and hopefully reach the required conclusion. It is important to notice that there is no one idealproof - a theorem can be established using different techniques and none of them will be better orworse (as long as they are all valid). For example, in Proofs from the book, we may findsix differentproofs of the infinity of primes (one of which is presented in section 7). Go ahead and master thetechniques - you might discover the passion for pure mathematics!



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We can divide the techniques presented in this document into two groups; direct proofs and indirectproofs. Direct proof assumes a given hypothesis, or any other known statement, and then logicallydeduces a conclusion.

Indirect proof, also called proof by contradiction, assumes the hypothesis (if given) together witha negation of a conclusion to reach the contradictory statement. It is often equivalent to proof by

contrapositive, though it is subtly different (see the examples). Both direct and indirect proofs mayalso include additional tools to reach the required conclusions, namely proof by cases or mathematicalinduction.

3.3 Implications and if and only if statements

If our hypothesis is about anything and everything and not about oneor more particular things, then our deductions constitute mathematics.Thus mathematics may be defined as the subject in which we never knowwhat we are talking about, nor whether what we are saying is true.Bertrand Russell

The formulaA =

B means A impliesB or ifA thenB , whereA and B are two statements.

SayingA = B indicates that whenever A is accepted, then we also must accept B . The importantpoint is that the direction of the implication should not be mixed! When A = B, then theargument goes from A to B, so ifA holds, then B does too (we cannot have A without B). On theother hand, when we have thatB is accepted, then it does nothave to happen that A is also accepted(so we can have B withoutA). This can be illustrated by the following example:

it is raining = it is cloudy.

Now, if the first statement is true (so it is raining), then we automatically accept that it is also cloudy.However, it does not work the other way round; the fact that it is cloudy does not imply the rain.Notice further, thataccepteddoes not mean true! We have that if it is raining, then it is cloudy andwe accept both statements, but we do not know whether they are actually true(we might have a nicesunny day!). Also, genuineness of the second statement does not give any information whether the first

statement is true or not. It may happen that the false statement will lead to the truth via a numberof implications!

If and only if, often abbreviated iff, is expressed mathematically A B and means that ifAholds, then B also holds and vice versa. To prove the theorems of such form, we must show theimplications in both directions, so the proof splits into two parts - showing that A B and thatB A. The proof of the statement

it is raining it is cloudy,

requires from us showing that whenever it is raining, then it is cloudy and showing that whenever itis cloudy, it is always raining.

Necessary and sufficient.

A B means that A is sufficient forB ;A B means that A is necessary forB ;A B means that A is bothnecessary and sufficient for B .



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4 Direct proof

4.1 Description of method

Direct proof is probably the easiest approach to establish the theorems, as it does not requireknowledge of any special techniques. The argument is constructed using a series of simple statements,where each one should follow directly from the previous one. It is important not to miss out any stepsas this may lead to a gap in reasoning. To prove the hypothesis, one may use axioms, as well as thepreviously established statements of different theorems. Propositions of the form

A = B

are shown to be valid by starting atA by writing down what the hypothesis means and consequentlyapproachingB using correct implications.

4.2 Hard parts?

it is tempting to skip simple steps, but in mathematics nothing is obvious - all steps of reasoningmust be included;

not enough explanations; I know what I mean is no good - the reader must know what youmean to be able to follow your argument;

it is hard to find a starting point to the proof of theorems, which seem obvious - we oftenforget about the axioms.

4.3 Examples

Below you will find the theorems from various areas of mathematics. Some of them will be newand techniques used not previously seen by the reader. To help with an understanding, the proofs arepreceded by the rough notes which should give a little introduction to the reasoning and show thethought process.

Theorem 4.1. Letn andm be integers. Then

i. ifn andm are both even, thenn+m is even,

ii. ifn andm are both odd, thenn+m is even,

iii. if one ofn andm is even and the other is odd, thenn+m is odd.

Rough notes. This is a warm-up theorem to make us comfortable with writing mathe-matical arguments. Start with the hypothesis, which tells you that both n and m areeven integers (for parti.). Use your knowledge about the even and odd numbers, writingthem in forms2k or2k+ 1 for some integerk .

Proof. i. Ifn and m are even, then there exist integers k and j such thatn = 2k and m = 2j. Then

n+m= 2k+ 2j = 2(k+j).

And sincek, j Z, (k+j) Z. n+m is even.ii. and iii. are left for a reader as an exercise.



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Theorem 4.2. Letn N, n >1. Suppose thatn is not prime = 2n 1 is not a prime.Rough notes. Notice that this statement gives us a starting point; we know what itmeans to be a prime, so it is reasonable to begin by writingn as a product of two natural

numbersn = a b.To find the next step, we have to play with the numbers so we receive the expressionof the required form.We are looking at 2ab 1 and we want to factorise this. We know the identity

tm 1 = (t 1)(1 +t+t2 + +tm1).Apply this identity with t = 2b andm = a to obtain

2ab 1 = (2b 1)(1 + 2b + 22b + + 2(a1)b).Always keep in mind where you are trying to get to - it is a useful advice here!

Proof. Since n is not a prime, a, b N such that n = a b, 1 < a,b < n. Let x = 2b 1 andy= 1 + 2

b

+ 2

2b

+ + 2(a1)b

. Then

xy= (2b 1)(1 + 2b + 22b + + 2(a1)b) (substituting for x and y)= 2b + 22b + 23b + + 2ab 1 2b 22b 23b 2(a1)b (multiplying out the brackets)

= 2ab 1 (taking away the similar items)= 2n 1. (as n= ab)

Now notice that since 1< b < n, we have that 1< 2b 1< 2n 1, so 1< x


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Proposition 4.4.x Z, 0 x= x 0 = 0

Rough notes. Striking theorem seen in the second year lecture! We all know it since our

early years, though now is the time to proove it! Again, we will refer to the axioms.

Proof.

x 0 = x (0 + 0) (because 0 = 0 + 0)= x 0 = x 0 +x 0 (by the distributivity)= 0 + (x 0) = x 0 +x 0 (by the existence of zero)= 0 = x 0 (by the cancellation)

Similarly, 0 = 0 x (try to prove it yourself!)

Theorem 4.5. n2+5n+1 asn .

Rough notes. Before we proceed, we need to recap the definitions.

Definition 4.1. A sequence (of real numbers) is a function fromN toR.

Definition 4.2. A sequence (an) of real numbers tends to infinity, if given any A >0,N N such thatan > A whenevern > N.

The above definitions are the key to proving the statement. We follow their structure,so we assume A being given and try to find N such that an > A whenever n > N.Proving statements of this form is not very hard, but requires practice to be able to get

the expression of required form. We will play with the fraction to make it smaller,which will prove that it tends to infinity.

Proof. Letan := n2+5n+1 and letA >0 be given. Observe that

an :=n2 + 5

n+ 1 n

2

n+ 1 (find an expression smaller than an by taking away 5 in the numerator)

n2

n+n (decrease an expression by increasing the denominator; holds as n N)

= n2

2n (addingns in the denominator)

= n2

(cancellingns in the numerator and denominator)

and n2 > A provided that n >2A.

So, let N be any natural number larger that 2A. Then ifn > N, we have an > n2

> N2

> A.Therefore,an tends to infinity.



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Lemma 4.6 (Gibbs Lemma). Assume(p1, . . . , pm)is probability distribution and let(r1, . . . , rm)

be such thatri> 0i andr

i=1

ri1. Theni

pi log 1

pii

pi log 1

ri.

Rough notes. You will probably not see this lemma until year 3, although a year 1 studentshould be able to prove it, as it requires only manipulations with logs and using simpleclaims. Notice, that we want to show the equivalent statement:

i

pi

log

1

pi log 1

ri

0.

Then there are just laws of logs and a simple claim that help us to arrive at the statement.

CLAIM: forx >0, ln xx 1.This is illustrated in the sketch below, however the formal proof is given in the appendix.

1 1 2 3 4 5 6

3

2

1

1

2

0

y= x 1

y = ln(x)

We now have all the tools needed to write proof of the lemma. See how it works step bystep and then check if you can do it yourself!

Proof. We want to show that i

pi log 1pi log 1ri 0.Write

i

pi

log

1

pi log 1

ri

=i

pi

log

1

pi+ log ri

(log(a1) = log a)

=i

pi

log

ripi

(log a + log b= log(ab))

= 1

ln m

i

pi

ln

ripi

(using the fact that log

ma= lna

lnm; mis a base of logarithm)

1

ln m i piri

pi 1 (by the claim)

= 1

ln m

i

ri pi

(multiplying out the brackets)

= 1

ln m

i

ri 1

i

pi

=1

(by the assumptions)

0.



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4.4 Fallacious proofs

Example 4.7. Study the sequence of sentences below and try to find what went wrong. You can findthe answer in the footnote1. We prove that 1 = 2.

a= b

= a2 =ab= a2 +a2 =a2 +ab= 2a2 =a2 +ab= 2a2 2ab= a2 +ab 2ab= 2a2 2ab= a2 ab= 2(a2 ab) = a2 ab= 2 = 1.

Example 4.8. This example will be similar to the previous one (again, we will prove that 2 = 1),although it does not contain the same mistake. Try to find what goes wrong here and again, thesolution is given at the bottom of the page2.

2 =2= 4 6 = 1 3= 4 6 +9

4= 1 3 +9

4

=

2 32

2

=

1 3

22

= 2 32

= 1 32

= 2 = 1.

Example 4.9. Below we will present the classic mistake of assuming true the statement which is yetto be proved. The task is to prove that the statement

2 +

6

Documents

2014 - University of Birmingham - Proofs and Math Reasoning