Axioms for the Set A of Integers

In this appendix, we state a collection of fundamental properties for the set of integers {... , -2, -1, 0, 1, 2, ...} that we have taken as axioms in the main body of the text. These properties provide the foundations for proving results in number theory.We begin with properties dealing with addition and multiplication. As usual, we denote the sum and product of a and b by a+ b and a· b, respectively. Following convention, we write ab for a· b.

• Closure: a + b and a· b are integers whenever a and b are integers.

• Commutative laws: a+ b = b+a and a·b = b ·a for all integers a and b.

• Associative laws: (a+ b) + c=a+ (b + c) and (a· b) ·c =a· (b ·c) for all integers a, b, and c.

• Distributive law: (a+ b) ·c =a· c+ b· c for all integers a, b, and c.

• Identity elements: a+ 0=a and a· 1 =a for all integers a.

• Additive inverse: For every integer a there is an integer solution x to the equation a+x = O; this integer xis called the additive inverse of a and is denoted by -a. By b - a, we mean b+ (-a).

• Cancellation law: If a, b, and care integers with a· c= b · c, c ¥=- 0, then a= b.

We can use these axioms and the usual properties of equality to establish additional properties of integers.An example illustrating how this is done follows. In the main body of the text, results that are easily proved from these axioms are used without comment.

Example A.1. To show that 0· a =0, begin with the equation 0+ 0 = O; this holds because 0 is an identity element for addition. Next, multiply both sides by a to obtain

(0+ 0) ·a= 0 · a. By the distributive law, the left-hand side of this equation equals

0 0 · 0 0 0 0 (0+ 0) · a = · a + a. Hence, ·a + · a = · a. Next, subtract ·a from both sides (which is the same as adding the inverse of 0·a). Using the associative law for addition and the fact that 0 is an additive identity element, the left-hand side becomes

0 0 0 0 0 - 0 0. 0·a+ (0·a - ·a) = ·a+ = ·a. The right-hand side becomes ·a ·a= We conclude that 0· a =0. ..,..

Ordering of integers is defined using the set of positive integers { 1, 2, 3, ...} . We have the following definition.

Definition. If a and b are integers, then a < b if b - a is a positive integer. If a < b, we also write b >a.

605 606 Axioms for the Set of Integers

Note that a is a positive integer if and only if a > 0.

The fundamental properties of ordering of integers follow.

• Closure for the positive integers: a + b and a · b are positive integers whenever a and b are positive integers.

• Trichotomy law: For every integer a, exactly one of the statements a> 0, a= 0, and a< 0 is true.

The set of integers is said to be an ordered set because it has a subset that is closed under addition and multiplication and because the trichotomy law holds forevery integer.

Basic properties of ordering of integers can now be proved using our axioms, as the following example shows. Throughout the text, we have used without proof properties of ordering that easily follow from our axioms.

Example A.2. Suppose that a, b, and c are integers with a< b and c> 0. We can show that ac < be. First, note that by the definition of a < b we have b - a > 0. Because the set of positive integers is closed under multiplication, c(b - a)> 0. Because c(b - a)= cb - ca, it follows that ca < cb. ..,..

We need one more property to complete our set of axioms.

• The well-ordering property: Every nonempty set of positive integers has a least ele­ ment.

We say that theset of positive integers is well ordered. On the other hand, the set of all integers is not well ordered, because there are sets of integers that do not have a smallest element (as the reader should verify). Note that the principle of mathematical induction discussed in Section 1.3 is a consequence of the set of axioms listed in this appendix. Sometimes, the principle of mathematical induction is taken as an axiom replacing the well-ordering property. When this is done, the well-ordering property follows as a consequence.

EXERCISES

1. Use the axioms for the set of integers to prove the following statements for all integers a, b,

andc. a) a· (b +c) =a· b+a· c c) a+ (b +c) = (c +a)+ b b) (a+b) 2 =a 2 +2ab + b2 d) (b - a)+ (c - b) +(a - c) =0

2. Use the axioms for the set of integers to prove the following statements for all integers a and b. a) (-1) · a = -a c) (-a)· (-1) =ab b) -(a· b) =a· (-b) d) -(a+ b) =(-a)+ (-b)

3. What is the value of -0? Give a reason for your answer. Axioms for the Set of Integers 607

4. Use the axioms for the set of integers to show that if a and bare integers with ab= 0, then a= 0 or b= 0.

5. Show that an integer ais positive if and only if a> 0.

6. Use the definition of the ordering of integers, and the properties of the set of positive integers, to prove the following statements for integers a, b, and c with a

a) a+ c be 2 3 b) a > 0 d) c <0

7. Show that if a, b, and c are integers with a> band b> c, then a> c.

* 8. Show that there is no positive integer that is less than 1.