Table Of ContentUnderstanding
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Proof e
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Explanation, Examples and p
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Solutions of Mathematic Proof
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Featuring
extended
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Edited by Tom Bennison and Edward Hall
Understanding Proof
Explanation, Examples and Solutions for
Mathematical Proof
Contributing Editors: Dr Tom Bennison and Dr Edward Hall
Picture Credits
Figure 4.1(a), Page 23, Public domain;
Figure 4.1(b), Page 23, Public domain;
Figure 4.2, Page 31, Shutterstock Image;
Figure 5.1, Page 47, Public domain;
Figure 6.1(a), Page 65, Public domain;
Figure 6.1(b), Page 65, Public domain;
Figure 7.1, Page 96, Public domain
Copyright (cid:2)c 2018 The Authors
Published by Tarquin
Book ISBN 978 1 91356 532 9 Ebook ISBN 978 1 91356 533 6
www.tarquingroup.com
This book may only be copied by licencees by agreement, under a Schools
Collective Licensing and on terms set by the Publisher.
First printing, November 2020
Contents
1 IntroductiontoProof ........................................... 5
2 ExploringMethodsofProof ..................................... 7
2.1 DirectProof 8
2.2 GraphicalProof 9
2.3 ProofbyContradiction 11
2.4 TheGeneralisedArithmetic-GeometricMeanInequality 11
3 MathematicalLanguage...................................... 13
3.1 StatementsandPredicates 13
3.2 Implication 17
4 DirectProof .................................................. 23
4.1 Examples of Direct Proof 24
4.2 Proof by Exhaustion 29
4.3 Proofs of Statements Involving the Positive Integers 32
4.4 Disproof by Counter Example 33
4.5 Direct Proofs From Higher Level Mathematics 35
5 IndirectProof ................................................. 47
5.1 ProofviatheContrapositive 48
5.2 ProofbyContradiction 50
6 ProofByInduction ............................................ 65
6.1 InductionforSeries 67
6.2 InductionforMatrices 73
6.3 InductionforDivisibility 77
6.4 InductionforInequalities 83
6.5 InductionforRecurrenceRelations 86
6.6 ApplicationsofProofbyInduction 88
7 ProofandApplicationsofPythagoras’Theorem ................ 96
7.1 ProvingPythagoras’Theorem 96
7.2 ProofsthatRequirePythagoras’Theorem 102
8 ProofsinCalculus ............................................ 112
8.1 DifferentialCalculus 112
8.2 IntegralCalculus 124
9 ProvingTrigonometricIdentities .............................. 129
9.1 TheStandardResults 129
9.2 ApplyingtheStandardIdentities 139
10 ProofsinStatisticsandProbability ............................. 149
10.1 DescriptiveStatistics 149
10.2 RandomVariables 152
11 WorkedSolutions ............................................ 158
1. Introduction to Proof
Throughoutourearlyschoolingwearetaughtthatmathematicsisaboutusingandmanip-
ulating numbers, perhaps with some application in mind and sometimes with seemingly
none. Of course, this is a part of what mathematics is about, but it is not the whole
story. A professional mathematician is not somebody who sits around thinking about
large numbers, for example. A mathematician is somebody who formulates conjectures
aboutobservationsandthentriestoshowiftheseconjecturesaretrueorfalse. Inthisway,
amathematician is similarto any other scientist, but ratherthan performing experiments
to prove their claims, mathematicians rely on logic and reason in their proofs. This is per-
haps the distinction between mathematics and other sciences, once a statement has been
proved mathematically, the logic is undeniable and the statement will remain true forever
more; the conjecture has become a theorem.
Perhaps the most familiar theorem of all is Pythagoras’ theorem that relates the lengths
ofthe sides ofatriangle. Pythagoras lived inthe6th century BC(although itis likely his
theorem was known well before this), but his theorem remains as true today as it did all
that time ago and it is just as useful, not just for its applications, but also for providing
the building blocks for the proof of more elaborate conjectures and theorems. In contrast,
other scientists build up a weight of evidence in support of a theory, but the theory is
never proven for definite. A classic example is Newton’s Laws, which provide a good
approximation the movement and interaction of objects under certain circumstances, but
theywereimproved andextendedby Leonhard Euler(ca. 1750)andthenagainbyAlbert
Einstein in the early 20th century when he developed his theory of Special Relativity. In
other sciences, theories evolves, whereas, in maths they remain for all time.
Itmightbetemptingtothinkthat,withthehistoryofmathematics stretchingbackmany
millennia, nearly everything has already been proved. This is actually far from the truth;
as time has gone on, more and more branches of mathematics have developed and more
and more questions have been posed that remain unanswered. In the year 2000, the Clay
6 Chapter1.IntroductiontoProof
Mathematics Institute offered prizes of $1 000 000 for anybody who could solve one of 7
outstanding mathematical problems (https://www.claymath.org). To this day (February
2018), only one of these problem has been solved, the Poincaré conjecture. The large sums
of money involved show how important these mathematical questions are and that it is
possible to earn a lot of money from mathematics.
This book is aimed at students who are just becoming acquainted with the concept of proof
and the rigour required when proving something. Clearly, there are very few results that an
student can hope to prove, but being exposed to the proofs behind well known results will
help a student gain a deeper insight and understanding of those results and develop their
capacity for logical thought. For some students who go on to study mathematics at a higher
level, this knowledge of how to formulate a proof will be essential and exposure to the
different methods of proof will be invaluable.
Each area of mathematics begins with a few results, which are either undeniable, impossible
to prove or just act as a starting point. These are called axioms; for example, the Peano
axioms define the arithmetic properties of natural numbers and include statements such as
‘0 is a natural number’ and ‘if x and y are natural numbers, then x = y implies y = x’. Once
we have a set of axioms, we can start to define other interesting objects and then begin to
prove new theorems about these new objects, which might in turn lead to more complex
definitions and further proof. A rich tapestry of mathematical ideas can quickly be built
in such a way. In order to prove new results, it is essential to have a deep understanding
of what has gone before and how proofs were set out. This is why studying simpler proofs
is essential for any mathematician. In some instances a proof can be formulaic and a
systematic approach will work, but, more often than not, proofs require some creativity.
In this book we cover the techniques of proof covered in upper school and lower College/
University mathematics syllabuses, including direct proof, proof by contradiction and proof by
induction. We shall also explain how these methods can be applied to other areas of
mathematics in the syllabus, for example, to geometry and trigonometry, calculus and
statistics. In doing so, we touch on proofs that are not core for a syllabus, but are an excellent
grounding for the student who intends to further their mathematical or scientific study.
2. Exploring Methods of Proof
As an introduction to methods of proof, we consider a number of different ways of ap-
proaching the proof of the arithmetic-geometric mean inequality. The methods presented
in this chapter will be explored in more depth throughout the rest of the book.
Before we can state the arithmetic-geometric mean inequality, we must define what the
arithmetic mean and geometric means are.
Definition2.1—ArithmeticMean
Consider two real numbers a and b, the arithmetic mean of these numbers is
a+b
m = .
a 2
Notice that the numbers a,m and b form an arithmetic sequence.
a
Definition2.2—GeometricMean
Consider two real numbers a≥0 and b≥, the geometric mean of these numbers is
√
m = ab.
g
In this case the name arises due to the numbers a,m and b forming a geometric se-
g
quence.
8 Chapter2.ExploringMethodsofProof
We now state the theorem.
Theorem2.3—TheArithmetic-GeometricMeanInequality
For two real numbers a≥0 and b≥0, the arithmetic-geometric mean inequality is:
√
1
(a+b)≥ ab. (2.1)
2
InteractiveActivity2.1—TheArithmetic-MeanInequality
Explore, visually, the arithmetic-geometric mean inequality using this Geogebra app.
2.1 DirectProof
Inthisdirectproof,westartwithknownfactsandperformanumberoflogicalsteps/deduction
until the we reach the desiredresult. Our starting point will bethe factthat (a−b)2 ≥0,
as any squared number is non-negative.
(a−b)2 ≥0,
⇒ a2−2ab+b2 ≥0,
⇒ a2+2ab+b2 ≥4ab,
⇒ (a+b)2 ≥4ab,
1
⇒ (a+b)2 ≥ab,
4
√
1
⇒ (a+b)≥ ab.
2
ProofTip
It is common to see “proofs” similar to the following.
√
1
(a+b)≥ ab
2
1
(a+b)2 ≥ab
4
(a+b)2 ≥4ab
a2+2ab+b2 ≥4ab
a2−2ab+b2 ≥0
(a−b)2 ≥0
Heretheresulthasbeenassumedandthenasequenceofdeductionstoatruestatement
has been made. This is not valid; to prove something in a deductive fashion we must
proceed to the result, not start from it. We must also provide some indication of how
we have gone from one line to the next, which has not been done in this case.
2.2 GraphicalProof 9
2.2 GraphicalProof
The figure below can be interpreted as a proof of (2.1). However, this requires some
understandingonbehalfofthereaderanditispreferabletowritesomewordsasguidance.
a b
ab
ab a
ab
ab b
Thelargersquare(ofsidelength a+b)hasanareagreaterthanthesumoftheareaofthe
four rectangles for any a(cid:6)=b. If a=b then the areas are the same. Hence,
√
1
(a+b)2 ≥4ab ⇒ (a+b)≥ ab,
2
proving the result.
The triangleshown belowcanbeusedtoprovide analternative proofof (2.1)usinggeom-
etry.
Q
1(a+b)
2
R 1(a−b) P
2
Since the triangle RPQ is right angled, we can apply Pythagoras’ Theorem to find the
