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. It is useful in proofs of properties of the integers, including in fermats method of. Chapter 6 proof by contradiction mcgill university. Propositional logic propositional resolution propositional theorem proving unification today were going to talk about resolution, which is a proof strategy. Employing the wellordering principle by our assumption, there is a natural number for which the predicate is false. Consider the following set which we define to be the set of natural numbers.
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. The wellordering principle n university of british. The wellordering theorem one of the greatest mathematical controversies of all time recall that the set of natural numbers with the order wellordered. We use a lemma here without proof, which is called the fundamental theorem of arithmatic fta. We then state what is known as the pigeonhole principle, and then we proceed to present an important method called mathematical induction. Principle of mathematical induction recall the following axiom for the set of integers. Let a be a xed integer, and let s be a set of integers such that i a is in s. We can show that the wellordering property, the principle of mathematical induction, and strong induction are all equivalent.
Some other less wellknown equivalents of the axiom of choice 3 3. In practice, induction and strong induction are more commonly used than well ordering. We start with a very important property of integers called the well ordering principle. First of all, it has been shown that if you want to believe the well ordering theorem, then it must be taken as an axiom. 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. In your textbook, there is a proof for how the wellordering principle implies the validity of mathematical induction. Note that it is usually used in a proof by contradiction. In a proof by contradiction, or indirect proof, you show that if a proposition were. 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. A nonempty subset s of r is wellordered if every nonempty subset of s has a smallest element. We actually have already taken the well ordering principle for granted in proving.
The well ordering principle of the natural numbers. Induction one of the most important properties of the set n 0, 1, 2. In your textbook, there is a proof for how the well ordering principle implies the validity of mathematical induction. Notice that 0 2s since x is clearly bounded below by 0.
The wellordering principle is a property of the positive integers which is equivalent to the statement of the principle of mathematical induction. Since l is the least element in s, l 1 62s, so pl 1 is true. The wellordering principle the wellordering principle is a concept which is equivalent to mathematical induction. If a is an integer larger than 1, then a can be written as a product of primes. Find materials for this course in the pages linked along the left. We can now prove theorem 67 using a proof by contradiction. Krantz1 february 5, 2007 amathematicianisamasterof criticalthinking,of analysis, andof deductive reasoning. Introduction to the theory of computation azadeh farzan winter 2010 monday, january 11, 2010. Assume there exists some positive integer that cannot be written as the product of primes. Proof methods such as proof by contradiction, or proof by induction, can lead to even more intricate loops and reversals in a mathematical argument. Proof by contradiction aka reductio ad absurdum, i.
Our inductive hypothesis is that for some n 0 we have n 2s. Proof techniques proof by contraposition 1b induction direct proof 1b proof by contradiction 1b simple induction 1c strong induction 1c trick. Axiom 71 well ordering principle every nonempty subset of nhas a. Let a be a xed integer, and let s be a set of integers such that 1.
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. Thus the wellordering principle, induction principle, and the induction principle are equally powerful. We show the wellordering principle implies the mathematical induction. Controversial results 10 acknowledgments 11 references 11 1. 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. The well ordering principle, iv zorns lemma, v tukeys lemma. For sake of contradiction, suppose that x has no minimum. Like induction, the wellordering principle can be used to prove that a. The theorem uses the well ordering principle or axiom. The resulting proof strategy is known as the smallest counterexample and is outlined below. By the wop on s, there is a smallest positive integer that cannot be. As we saw in class, the well ordering principle is equivalent to the principle of mathematical induction. Discrete structures homework assignment 3 solutions exercise 1 20 points. In this paper we will look at the axiom of choice and some of the.
Also, is there a context where one can simply take this principle as an axiom, and not have to prove it. Mathematics for computer science open data structures. We want to establish that s n by the wellordering principle. This is nice, but what we can do with it ends up being authentically excellent. The well ordering principle 61 use a proof by contradiction and assume that cis nonempty. Hence, we shall regard the principle of well ordering as an axiom. Next, argue that there is no smallest element of s by doing a proof by contradiction. In fact, we cannot prove the principle of well ordering with just the familiar properties that the natural numbers satisfy under addition and multiplication. First, well look at it in the propositional case, then in the first order case. Since tis closed, there is a least upper bound function on chains in t, g. To prove that pn is true for all positive integers n. Like induction, the well ordering principle can be used to prove that a collection of statements indexed by the natural numbers is true.
These skills travel well, and can be applied in a large variety of situationsand in many di. Proving the socalled well ordering principle stack exchange. This principle can be taken as an axiom on integers and it will be the key to proving many theorems. I get the sense there is something wrong here, but i cant seem to define. Conclude that the principles of induction, strong induction, and wellordering are. Feb 29, 2020 in this section, we present three basic tools that will often be used in proving properties of the integers. Every nonempty set s s s of nonnegative integers contains a least element. Suppose now for the sake of contradiction that there is a pair x. To prove p to prove p, assume p and find a contradiction q such that p q is true. Then, the book moves on to standard proof techniques.
The well ordering principle and mathematical induction. Induction one of the most important properties usually taken to be an axiom of the set n f1. 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. Every nonempty subset of \\mathbbn\ has a smallest element. See the course calendar for the full office hours schedule. Consider proving the following summation to be true for all positive integers n. Then there is a least element l in s by the well ordering principle. Wellordering principle schedule this week, you should read mcs chapter 2 and mcs chapter 3 at least through the end of section 3. The well ordering principle again we assume the axiom of choice. Here are several examples of properties of the integers which can be proved using the well ordering principle. This result is called the well ordering principle, which we will take as an axiom. How to prove the well ordering principle using induction.
Assume for proof by contradiction that c is nonempty. We can show that the well ordering property, the principle of mathematical induction, and strong induction are all equivalent. Reach a contradiction somehowoften by showing that p. The well ordering principle the well ordering principle is a concept which is equivalent to mathematical induction. We show the well ordering principle implies the mathematical induction. 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. Discrete structures homework assignment 3 solutions. 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.
Reach a contradiction somehowoften by showing how to use nto. Strengthening the original statement 1c proof by cases well ordering principle 1d. Equivalence between the axiom of choice and the claim that every vector space has a basis 5 3. The wellordering principle of the natural numbers mathonline. That is, suppose that we need to prove that whenever the statement p holds true, the statement q holds true as well. Assume for the sake of contradiction that s is nonempty, so by the wellordering principle it has a least element l. This new method is not limited to proving just conditional statements it can be used to prove any kind of statement whatsoever. That is, the validity of each of these three proof techniques implies the validity of the other two techniques. Let s be the set of positive integers that do not have a prime factorization. This week, you should read mcs chapter 2 and mcs chapter 3 at least through the end of section 3. These techniques will be useful in more advanced mathematics courses, as well as courses in statistics, computers science, and other areas. The axiom of choice can be proven from the well ordering theorem as follows. If s n is a subset of the natural numbers such that i1 2s, and ii whenever k 2s, then k. In fact, looking back, we took the well ordering principle for granted in prov ing that v 2 is irrational.
The wellordering principle is a property of the positive integers which is. But in fact, it provides one of the most important proof rules in discrete mathematics. 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. We show the wellordering principle implies the math ematical induction. The principle of strong mathematical induction is equivalent to both the well ordering principle and the principle of mathematical induction.
Proofs the well ordering principle duke computer science. It will actually take two lectures to get all the way through this. Then by thewellordering principle there is a least element m 2 nns. 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. By the well ordering principle, there will be a smallest element, n, in c. Proofs proof by contradiction proof by construction jack sees jill, who has just come in from outdoors proof by induction dry. In general, a set such as n with some order wellordered if any nonempty subset has a least element. To make a choice function for a collection of nonempty sets, e, take the union of the sets in e and call it x. In order to use induction and we will need strong induction exercise 10. Well ordering, division, and the euclidean algorithm. It can also be stated for all sets, not just sets of integers and is related to zorns lemma and the axiom of choice. Assume for the sake of contradiction that b is nonempty.
Furthermore, this factorization is unique except for the order of the factors. Assuming the principle of mathematical induction as an axiom, the wellordering property of n holds. To conclude, since each principle can be proved from the other, any problem solvable with one can also be solved by the other. Both parts of the proof will use the well ordering principle for the set of natural numbers. A nonempty subset s of r is well ordered if every nonempty subset of s has a smallest element. Actually, the well ordering principle could also be proven using the principle of mathematical induction. The history and concept of mathematical proof steven g. The axiom of choice, zorns lemma, and the well ordering principle 3 proof. If i remember correctly, mathematical induction uses the well ordering principle as a proof for it.
Then by thewell ordering principle there is a least element m 2 nns. Chapter 6 proof by contradiction we now introduce a third method of proof, called proof by contra diction. I get the sense there is something wrong here, but i cant seem to define exactly what. My first proof that uses the wellordering principle very. Verifying the above by contradiction we can lead like. 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 every nonempty subset of the natural numbers contains a least element. Proofs using well ordering and induction of the irrationality of square root of 2. Using the well ordering principle in proofs let pn be a statement involving a natural number n. Strong induction and well ordering york university. The well ordering principle a least element exist in any non empty set of positive integers. We want to establish that s n by the well ordering principle. Thus, we can use the axiom of choice to choose one pair a,y 2 y for every y 2.
N is a subset of the natural numbers such that i 0. To use the a descent proof, we need to work with natural numbers, i. So, it seems a bit circular to use proof by mathematical induction to prove. Unfortunately for him, his proof was soon shown to be fatally awed and the question still open.
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