Here are several examples of properties of the integers which can be proved using the well ordering principle. Introduction to the theory of computation azadeh farzan winter 2010 monday, january 11, 2010. To prove p to prove p, assume p and find a contradiction q such that p q is true. Unfortunately, not all proposed proofs of a statement in mathematics are actually correct, and so some e ort needs to be put into veri cation of such a proposed proof. Since l is the least element in s, l 1 62s, so pl 1 is true. Hence, we shall regard the principle of well ordering as an axiom. Strengthening the original statement 1c proof by cases well ordering principle 1d. Then, the book moves on to standard proof techniques.
Let a be a xed integer, and let s be a set of integers such that 1. Like induction, the wellordering principle can be used to prove that a. It is useful in proofs of properties of the integers, including in fermats method of. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. Thus the wellordering principle, induction principle, and the induction principle are equally powerful. That is, suppose that we need to prove that whenever the statement p holds true, the statement q holds true as well. It will actually take two lectures to get all the way through this. Assume there exists some positive integer that cannot be written as the product of primes. In peano arithmetic, second order arithmetic and related systems, and indeed in most not necessarily formal mathematical treatments of the wellordering principle, the principle is derived from the principle of mathematical induction, which is itself taken as basic.
The resulting proof strategy is known as the smallest counterexample and is outlined below. Krantz1 february 5, 2007 amathematicianisamasterof criticalthinking,of analysis, andof deductive reasoning. The wellordering principle n university of british. That is, the validity of each of these three proof techniques implies the validity of the other two techniques.
Since tis closed, there is a least upper bound function on chains in t, g. Next, argue that there is no smallest element of s by doing a proof by contradiction. The well ordering principle again we assume the axiom of choice. The axiom of choice, zorns lemma, and the well ordering principle 3 proof. We can show that the well ordering property, the principle of mathematical induction, and strong induction are all equivalent. By the well ordering principle, there will be a smallest element, n, in c. Induction one of the most important properties usually taken to be an axiom of the set n f1. I get the sense there is something wrong here, but i cant seem to define. Induction, strong induction, and well ordering are logically equivalent, so the best choice for a particular application is the one that you think gives the clearest proof. In order to use induction and we will need strong induction exercise 10. By the wop on s, there is a smallest positive integer that cannot be. A nonempty subset s of r is well ordered if every nonempty subset of s has a smallest element.
Notice that 0 2s since x is clearly bounded below by 0. For sake of contradiction, suppose that x has no minimum. This week, you should read mcs chapter 2 and mcs chapter 3 at least through the end of section 3. The well ordering principle every nonempty subset of the natural numbers contains a least element. But in fact, it provides one of the most important proof rules in discrete mathematics. The well ordering principle, iv zorns lemma, v tukeys lemma. Mathematics for computer science open data structures. Controversial results 10 acknowledgments 11 references 11 1.
These skills travel well, and can be applied in a large variety of situationsand in many di. Every nonempty set s s s of nonnegative integers contains a least element. The well ordering principle the well ordering principle is a concept which is equivalent to mathematical induction. The axiom of choice and its implications 3 words, for every distinct y,z 2. We then state what is known as the pigeonhole principle, and then we proceed to present an important method called mathematical induction. We show the well ordering principle implies the mathematical induction. Proof techniques proof by contraposition 1b induction direct proof 1b proof by contradiction 1b simple induction 1c strong induction 1c trick.
Every integer greater than 1 is either prime itself or can be written as a unique product of prime numbers apart from the order of the primes. Suppose now for the sake of contradiction that there is a pair x. In practice, induction and strong induction are more commonly used than well ordering. The axiom of choice and its implications kevin barnum abstract. Employing the wellordering principle by our assumption, there is a natural number for which the predicate is false. We can now prove theorem 67 using a proof by contradiction. Induction one of the most important properties of the set n 0, 1, 2. If a is an integer larger than 1, then a can be written as a product of primes. Furthermore, this factorization is unique except for the order of the factors. In your textbook, there is a proof for how the wellordering principle implies the validity of mathematical induction. There is of course one well known, named in nite set of numbers which is well ordered, and this will be the crux of what we do henceforth. Use the well ordering property to prove if a is an integer and d is a positive integer, then there are unique integers q and r with 0 r proof by well ordering. Strong induction and well ordering york university.
Our inductive hypothesis is that for some n 0 we have n 2s. In fact, we cannot prove the principle of well ordering with just the familiar properties that the natural numbers satisfy under addition and multiplication. We can show that the wellordering property, the principle of mathematical induction, and strong induction are all equivalent. So, it seems a bit circular to use proof by mathematical induction to prove. The wellordering theorem one of the greatest mathematical controversies of all time recall that the set of natural numbers with the order wellordered. Discrete structures homework assignment 3 solutions exercise 1 20 points. Proving the socalled well ordering principle stack exchange.
Also, is there a context where one can simply take this principle as an axiom, and not have to prove it. Wellordering principle schedule this week, you should read mcs chapter 2 and mcs chapter 3 at least through the end of section 3. Reach a contradiction somehowoften by showing how to use nto. The well ordering principle a least element exist in any non empty set of positive integers. We begin our look through abstract algebra with a rather simple theorem regarding the set of natural numbers known as the well ordering principle of the natural numbers. See the course calendar for the full office hours schedule. Assuming the principle of mathematical induction as an axiom, the wellordering property of n holds. The well ordering principle of the natural numbers. We show the wellordering principle implies the math ematical induction. Principle of mathematical induction recall the following axiom for the set of integers. To prove that pn is true for all positive integers n. Consider proving the following summation to be true for all positive integers n. If i remember correctly, mathematical induction uses the well ordering principle as a proof for it.
The wellordering principle the wellordering principle is a concept which is equivalent to mathematical induction. Then there is a least element l in s by the well ordering principle. Every nonempty subset of \\mathbbn\ has a smallest element. We want to establish that s n by the well ordering principle. A nonempty subset s of r is wellordered if every nonempty subset of s has a smallest element. Conclude that the principles of induction, strong induction, and wellordering are. Chapter 6 proof by contradiction mcgill university.
Axiom 71 well ordering principle every nonempty subset of nhas a. Discrete structures homework assignment 3 solutions. In a proof by contradiction, or indirect proof, you show that if a proposition were. It can also be stated for all sets, not just sets of integers and is related to zorns lemma and the axiom of choice. Every nonempty set of positive integers contains a smallest member. Unfortunately for him, his proof was soon shown to be fatally awed and the question still open. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. In your textbook, there is a proof for how the well ordering principle implies the validity of mathematical induction. First i will show you an example of a proof that utilizes the well ordering principle, then i will show how the wellordering principle implies mathematical induction. Well ordering, division, and the euclidean algorithm. We actually have already taken the well ordering principle for granted in proving. Verifying the above by contradiction we can lead like.
The wellordering principle is a property of the positive integers which is. To make a choice function for a collection of nonempty sets, e, take the union of the sets in e and call it x. The theorem uses the well ordering principle or axiom. I get the sense there is something wrong here, but i cant seem to define exactly what. A proof using the principle of mathematical induction noting that a proof using the wellordering principle can usually be converted to a proof using the principle of mathematical induction, and vice versa, i was pleasantly surprised that i could easily construct the following proof. In peano arithmetic, secondorder arithmetic and related systems, and indeed in most not necessarily formal mathematical treatments of the well ordering principle, the principle is derived from the principle of mathematical induction, which is itself taken as basic. We start with a very important property of integers called the well ordering principle. We show the wellordering principle implies the mathematical induction. Some other less wellknown equivalents of the axiom of choice 3 3.
We want to establish that s n by the wellordering principle. In general, a set such as n with some order wellordered if any nonempty subset has a least element. This is nice, but what we can do with it ends up being authentically excellent. To conclude, since each principle can be proved from the other, any problem solvable with one can also be solved by the other. By the axiom of choice, there exists a function that assigns to each proper subset sof t an element of t s. The well ordering principle 61 use a proof by contradiction and assume that cis nonempty. To use the a descent proof, we need to work with natural numbers, i. Let a be a xed integer, and let s be a set of integers such that i a is in s. The wellordering principle of the natural numbers mathonline. Let s be the set of positive integers that do not have a prime factorization. We use a lemma here without proof, which is called the fundamental theorem of arithmatic fta. Feb 29, 2020 in this section, we present three basic tools that will often be used in proving properties of the integers.
Reach a contradiction somehowoften by showing that p. Principle of mathematical induction for predicates. Proof methods such as proof by contradiction, or proof by induction, can lead to even more intricate loops and reversals in a mathematical argument. These techniques will be useful in more advanced mathematics courses, as well as courses in statistics, computers science, and other areas. Actually, the well ordering principle could also be proven using the principle of mathematical induction. Find materials for this course in the pages linked along the left. Thus, we can use the axiom of choice to choose one pair a,y 2 y for every y 2. This principle can be taken as an axiom on integers and it will be the key to proving many theorems. First, well look at it in the propositional case, then in the first order case.
As we saw in class, the well ordering principle is equivalent to the principle of mathematical induction. The history and concept of mathematical proof steven g. The wellordering principle is a property of the positive integers which is equivalent to the statement of the principle of mathematical induction. Then by thewellordering principle there is a least element m 2 nns. In fact, looking back, we took the well ordering principle for granted in prov ing that v 2 is irrational. N is a subset of the natural numbers such that i 0. Sep 25, 2017 well ordering principle proof examples, well ordering principle proof by induction, well ordering principle problems, well ordering principle axiom of choice, well ordering principle pdf, well. Well ordering axiom for the integers if b is a nonempty subset of z which is bounded below, that is, there exists an n 2 z such that n b for. Noting that a proof using the well ordering principle can usually be converted to a proof using the principle of mathematical induction, and vice versa, i was pleasantly surprised that i could easily construct the following proof. Consider the following set which we define to be the set of natural numbers. How to prove the well ordering principle using induction. In this paper we will look at the axiom of choice and some of the. Both parts of the proof will use the well ordering principle for the set of natural numbers.
Proof by contradiction aka reductio ad absurdum, i. Assume for the sake of contradiction that b is nonempty. Note that it is usually used in a proof by contradiction. The axiom of choice can be proven from the well ordering theorem as follows. Proofs proof by contradiction proof by construction jack sees jill, who has just come in from outdoors proof by induction dry. Using the well ordering principle in proofs let pn be a statement involving a natural number n. Assume for the sake of contradiction that s is nonempty, so by the wellordering principle it has a least element l. Like induction, the well ordering principle can be used to prove that a collection of statements indexed by the natural numbers is true. The well ordering principle and mathematical induction.
Propositional logic propositional resolution propositional theorem proving unification today were going to talk about resolution, which is a proof strategy. Proofs using well ordering and induction of the irrationality of square root of 2. Equivalence between the axiom of choice and the claim that every vector space has a basis 5 3. This result is called the well ordering principle, which we will take as an axiom.