PROBLEM SOLVING. (2n +1) 3 =8n 3 +12n 2 +6n +1=2(4n 3 +6n 2 +3n)+1.

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1 CONTENTS PREFACE PROBLEM SOLVING. PROOF BY CONTRADICTION: GENERAL APPROACH 5. INDUCTION. ROTATIONS 4 4. BARYCENTRIC COORDINATES AND ORIENTED (SIGNED) AREA 6 5. INVARIANTS 9 vii

2 PROBLEM SOLVING The Art of Problem Solving In this book we will focus on certain methods of problem solution. A good problem solver must first of all have a clear understanding of what constitutes mathematical solution. There are many ways of constructing mathematical solutions, but there is no systematic way of teaching the variety of methods which are used in mathematics. First, however, it is necessary to have a perspective of some of the most fundamental methods of proof. Some knowledge of logic will also be useful. Methods of The most intuitive method of proof is a direct proof. In this form of proof all of the information given is considered and by a sequence of permitted steps we work towards the result. There is no systematic way of developing the sequence of logical steps in a direct proof. This will develop with experience, and will benefit from the originality of the student. An example of a direct proof is as follows. Example Prove that the cube of an odd number is also odd. We are given an odd number. Thus it can be written as n +, wheren is an integer. Then its cube is (n +) =8n +n +6n +=(4n +6n +n)+. Since 4n +6n +n must be an integer, it becomes even whenmultiplied by and odd when is added. So the cube is odd. You will note how we started with the information given and led ourselves to the result by a sequence of logical arguments. As is often the case much of our argument involved algebraic manipulation and rearrangement. Direct proof is a fundamental, intuitive method of proof. In this book we will look at other methods of proof per se, as well as general methods of problem solving. In the first chapter wewill look at a very useful method of proof: proof by contradiction.

3 . PROOF BY CONTRADICTION: GENERAL APPROACH by Contradiction Whereas direct proof is an intuitive method which is verypopular, itcan become more complicated when the problem is more complicated. Often we find that the method of proof by contradiction provides a simple proof structure which creates an environment in which the argument can follow easily. In this method we commence by assuming that the result that we have to prove is false. We then show by a sequence of logical steps that this causes one of the pieces of given information to be false, i.e. we obtain a contradiction. Why does this type of argument constitute mathematical proof? There are many ways of looking at it. Some people do not need to be convinced. Most textbooks and mathematics courses contain several proofs by contradiction without ever seeing the need to justify the strategy. A formal course in logic will always justify the strategy, on the other hand, by highly rigorous argument. We will justify by the use of Venn diagrams, i.e. diagrams which provide aqualitatitive representation of sets (in our case truth sets, the sets of elements from a universal set whose substitution into an open sentence results in a true proposition). It would be a diversion for us here to go into formal discussion of these concepts in logic. We hope that the sense will, however, be fairly obvious. If two sets A and B are mutually disjoint (i.e. have no elements in common) they can be represented in a Venn diagram by completely separate circles, e.g... A B (Note that in drawing a circle we are representing the whole disc, not just the boundary.)

4 6. by Contradiction: General Approach If the sets have some elements in common, we say that they have a non-empty intersection, and can be represented by the following Venn diagram, in which the circles intersect... A B Now let us suppose that all elements of set B are contained in set A (but A may have some extra elements which are not in B). This could be described by B A.. We note that if an element is in B it is also in A. FromtheVenn Diagram we observe that this is equivalent to saying that if an element is outside A it is also outside B. In logical terms this translates to saying that if statement P implies statement Q it is always true to say that if statement Q is false then statement P is false (known as the contrapositive of the original statement). This, then, is the basis of the strategy in proof by contradiction. If we are trying toprovethatstatementp (the statement of the given conditions) implies statement Q (the result we are to prove) it is just as good to provethatifstatement Q is false (i.e. assume the contrary to the result we are to prove) we obtain a contradiction (i.e. show that this requires the giveninformation (P ), or part thereof, to be false). There are also many ways of illustrating this concept in a sentence. For example we could say that Bulgaria is a country in Europe. This is logically equivalent to saying that if a country is not in Europe then it is not Bulgaria (convoluted as it may seem).

5 . by Contradiction: General Approach 7 Two of the best known results in mathematics, and both results published by Euclid of Alexandria, about 00BC, in his Elemenrs, discussed below, are normally proved using contradiction. The first result is from Number Theory. THEOREM There are infinitely many prime numbers. Suppose the contrary, i.e. that there is only a finite number of primes. Let these be p,p,,p n for finite integer n. Considerthe number p p p n +. This is different from all the given primes (it is clearly larger than any of them) and it is not divisible by any of the primes p,p,,p n.thisnew number must be either an additional prime number, or a number divisible by an additional prime number, contradicting the original assumption. The next result, stating that isanirrational number, isequally wellknown. THEOREM The number isirrational, i.e. it cannot be expressed in the form p/q, where p and q are integers. Suppose the contrary, i.e. that p = q. Without loss of generality, we can assume that p and q have no common factors. Then = p q, i.e. p =q. Thus p is divisible by, and hence p is divisible by. Write p =r, wherer is an integer. Then 4r =q,orq =r.thus q is divisible by, and hence q is divisible by. So p and q are both divisible by. This contradicts our original assumption, completing the proof.

6 8. by Contradiction: General Approach Note: This theorem caused a sensation when it was first discovered (perhaps between 50 and 500 B.C. The proof given here is almost certainly the first proof, and was probably due to Pythagoras himself. There are many other theorems which illustrate these proof structures. Two of the most important are in the theory of real numbers. A set of numbers (possibly infinite) is said to be countable if its elements can be arranged in order, as members of a sequence, and uncountable otherwise. If finite, a set is trivially countable. The fact that the rational numbers are countable can be most commonly proved by a direct proof, in which weshowdirectly how a sequence including all rational numbers can be constructed. THEOREM The rational numbers are countable. Consider the set Q +, i.e. the set of all positive rational numbers. We can certainly arrange them all in the array Arrange the members of Q + in the sequence given by following the arrows {,,,,,,.} Remove every term which equals a preceding term (e.g.,whichequals ). What remains gives the positive rationals arranged in a sequence without repetition. p,p,p,