Bridge to Abstract Mathematics: Mathematical Proof and Structures

10.1 AN AXlOMATlZATlON FOR THE SYSTEM OF POSITIVE INTEGERS 339

must show that p(q + 1) # 1. Now p(q + 1) = pq + p = p + pq = p +

o(x) = a(p + x), where p + x E N, by additive closure. Hence p(q + 1) E

im (a). Since 1 4 im (a), we conclude 1 # p(q + I), as desired. 0

ORDERING PROPERTIES OF N

We now consider properties of N having to do with the ordering of positive
integers. Our starting point is the following definition.

DEFINITION 3

Let a, b E N. We say that a < b (a is less than b) if and only if there exists c E N

such that a + c = b.

A number of facts related to Definition 3 present themselves immediately.
We collect some of them in the following theorem, whose proof is left to
you in Exercise 5(a).

THEOREM 8

Let a, b, c E N. Then:

(a) a < 4a)

(b) a<a+b

(c) If a < b and b < c, then a < c (transitivity)

(d) If a < b, then a # b

(e) If a # 1, then 1 <a

(f) If a < 6, then there exists a unique x E N such that a + x = b

(g) If a < b, then a + c < b+ c

(h) If a < b, then ac < bc

The uniqueness in part (I) of he or em 8 is suggestive of the subtraction
operation, as defined in Definition 4.

DEFINITION 4

If a, b E N with a < b, we denote by b - a (b minus a) the unique positive integer

x such that a + x = b.

We emphasize-that b - a is defined in N if and only if a < b. We collect
other properties in Theorem 9.

THEOREM 9

Let a, b, c E N with a < b < c. Then:

(a) a+(b-a)=b

(6) (c - b) + a = c - (6 - a)

(c) (c + b) - a = c + (b - a)

(d) c(b - a) = cb - ca