346 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10
one (Verify!) and is not onto. In fact, an equivalence class [(p, q)] in Z is
in the image of this mapping if and only if p > q. Finally, this mapping
preserves the operations of addition and multiplication, that is, [n,] +
[n,] = [n, + n,] and [n,][n,] = [nln2] for any positive integers n, and
n,. Stated differently, if n, is associated with [(p,, q,)] and n, is associated
with [(p,, q,)], then n, + n, is associated with [(p, + p,, q, + q,)] and
n,n2 is associated with [(p,p, + q,q,, p,q2 + q,p,)]. The proofs of these
facts are left to you in Exercise 2(c).
Thus we may think of Z as containing a proper subset; let us call it
Z+ = {[(p, q)]lp E N, q E N, p > q}, which is, for all intents and purposes,
identical to N. (If you study abstract algebra later, you will come to rec-
ognize that we have established here an isomorphism between N and Z'.)
It is in this technical sense that N may be regarded as a subset of the system
Z that we have constructed.
As to properties of Z, you should verify that a number of properties of
N, proved for N in Article 10.1 and listed in Exercise 3, "carry over" to Z.
In addition, Z has desirable properties that N fails to possess, including
Theorem 3.
THEOREM 3
There exists a unique integer a having the property that x + a = a + x = x
for all x E 2. We denote this integer by the symbol "0" and call it the zero
element or additive identity of Z.
To each x E Z, there corresponds a uniquely determined y E Z such that
x + y = y + x = 0. We denote this integer by the symbol -x and call it
the additive inverse of x.(a) Let a = [(I, I)], and let x = [(p, q)] be an arbitrary integer.
Then x + a = [(I + p, 1 + q)], which clearly equals [(p, q)], so that
x + a = x, as desired. It is left to you to establish the facts that a + x = x
and that a is unique.
(b) Let x = [(p, q)] be an arbitrary integer. Consider the integer
y = [(q, p)] and note that x + y = [(p + q, q + p)] = [(I, l)] = 0. Again,
you should verify that y + x = 0 and that this y is the only integer
satisfying x + y = y + x = 0. 0With Exercise 3 and Theorem 3, we see that the integers satisfy all but
possibly one of the 11 field axioms. It is worth noting here that because
(Z, +, -) is an algebraic structure with two operations satisfying field axioms
1 through 7 and 11, it is an example of what algebraists call a ring. In fact,
because Z satisfies Axiom 8, it is a commutative ring, and since it satisfies
Axiom 9, it is a ring with unity. We will note an additional, and highly
important, property of Z, in Theorem 5(b). Also, we will address the ques-
tion of Axiom 10, multiplicative inverses in Z, following the proof of Theo-
rem 6. You will recall that, given a, b E N, the equation a + x = b does not