1.7 Proof Methods and Strategy

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1 1.7 Proof Methods and Strategy Proof Strategies Finding proofs can be difficult. Usual process of proving: Replace terms by their definitions Analyze what the hypothesis and the conclusion mean Try to prove the result by using available methods of proof. FORWARD AND BACKWARD REASONING Independent of your choice of proof, you need a starting point for your proof. If the proof starts with the premises and constructs steps leading to the conclusion, then this type of reasoning is called forward reasoning. When forward reasoning is not working, then backward reasoning can be used. Example14. Given x, y R^{+} with x y, their arithmetic mean is (x+y)/2 and their geometric mean is (xy)^{1/2}. Can we prove that (x+y)/2>(xy)^{1/2}? Solution: Since it is not obvious how to reach the conclusion, we can work backwards-start working on the conclusion and then see if all the steps can be reverted. If the steps are revertible (equivalent), we can construct a forward reasoning. ADAPTING EXISTING PROOFS Existing proofs can be adapted to prove a new result or some of the ideas used in existing proofs can be helpful. Looking for Counterexamples A conjuncture ends up either with a proof or with a counterexample. In any case looking for counterexamples is extremely important. It provides insights into the problems. Example17. Every positive integer is the sum of the squares of two integers is proven to be false by a counterexample. Now what about Every positive integer is the sum of the squares of three integers? Solution: the first thing we would like to do is to check it for some integers to see if it is going to tend to be true or if we see one counterexample. (Check it until n=7). 1

2 What about Every positive integer is the sum of the squares of four integers? It is true. Proof Strategy in Action Usual mathematics texts formally present theorems and their proofs. Such presentations do not reveal the discovery process in mathematics. This process begins with exploring concepts and examples, asking questions, formulating conjunctures, and attempting settle these conjunctures either by proof or by counterexample. Conjunctures are formulated on the basis of many types of possible evidence. Once a conjuncture is formulated, people will look for a proof or for a counterexample. The Role of Open Problems Famous unsolved problems lead to advances in mathematics. Quote: As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Theorem 1 Fermat s Last Theorem The equation x^{n}+ y^{n}=z^{n} has no solutions in x, y, z N with xyz 0 and n N with n>2. Remark: The equation x^{2}+ y^{2}=z^{2} has infinitely many solutions in x, y, z N with xyz 0. These solutions are called Pythagorean triples and correspond to the lengths of the sides of right triangles with integer length. Andrew Wiles has proven the Fermat s Last Theorem using the recently developed theory of elliptic curves of number theory. Example 23. The 3x+1 Conjuncture Let T be the transformation that sends an even integer x to x/2 and an odd integer x to 3x+1. The 3x+1 Conjuncture states that all positive integers x, when we repeatedly apply the transformation T, we will eventually reach the integer 1. For example x=13. This conjuncture has been verified for all integers x up to 5.6x10^{13}. 2

3 Additional Proof Methods In this chapter we have introduced the basic methods used in proofs. Note that we have not given a procedure that can be used to prove theorems in mathematics. Such a procedure is not existent! Chapter 2 Basic Structures: Sets, Functions, Sequences, and Sums 2.1 Sets Introduction In this section we study the set fundamental discrete structure on which all other discrete structures are built. Definition 1 A set is an unordered collection of objects. We will use Cantor s original version of set theory naive set theory. Definition 2 The objects in a set are called the elements, or members, of the set. A set is said to contain its elements. Notation: a A denotes that a is an element of the set A. Another way to describe a set is to use set builder notation. We characterize all elements in the set by stating the property they must have to be members. Important sets: N={0, 1, 2, 3, }- the set of natural numbers, Z={,-3,-2,-1, 0, 1, 2, 3, }- the set of integers Z^{+}={1, 2, 3, }- the set of positive integers, Q={x x=p/q and p, q Z with q 0} - the set of rational numbers, Q^{+}={x x=p/q and p, q Z^{+} } - the set of positive rational numbers, R, the set of real numbers. 3

4 Remark: Note that the concept of a datatype, or a type in computer science is built upon the concept of a set. Definition 3 Two sets are equal if and only if they have the same elements. That is, if A and B are sets, then A and B are equal if and only if x(x A x B). We write A=B, if A and B are equal sets. Example 6. The sets {1,3,5} and {5,3,1} are equal- they have the same elements. Definition 4 The set A is said to be a subset of B if and only if every element of A is also an element of B. We use the notation A B to indicate that A is a subset of the set B. Note that A B x(x A x B). Theorem 1 For every set S, (i) S and (ii) S S. Proof is straightforward by the Definition 4. When we wish to emphasize that a set A is a subset of B but that A B, we write A B and we say that A is a proper subset (real subset) of B. Note that A B x(x A x B) x(x B x A); A=B (A B) (B A) or "x(x A«x B) ( x(x A x B)) ( x(x B x A)). Sets may have other sets as members. A={, {a}, {b}, {a, b}} and B={x x is a subset of the set {a, b}}. Obviously A=B. 4

5 Definition 5-6 Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S denoted by S. A set is said to be infinite if it is not finite. Example 10. Let S be the set of letters in the English alphabet. Then S =26. Example 12. The set of positive integers Z^{+}={1, 2, 3, } is infinite. The Power Set Definition 7 Given a set S, the power set of S is the set of all subsets of the set S. The power set of S denoted by P(S). Example 13 What is the power set of the set {0,1,2}? Note that the empty set and the set itself are members of this set of subsets. Example 14 What is the power set of the empty set? What is the power set of the set { }? Solution: The empty set has exactly one subset. Consequently P( )(={, })={ }. The set { } has exactly two subsets, the empty set { } and itself { }. Therefore P({ })={, { }}. 5