4 Chapter 1 • The Real Number System
Some number systems are fields, but many are not. The following exercises
will help you see the difference.
E XERCISE SET 1.1- A
In Exercises 1- 10, a specific set is given and two operations on that set
are specified. On the basis of your previous experience, tell which of the field
axioms are satisfied in the given mathematical system. Justify your answers
briefly.
- The set N of n atural numbers,^2 with ordinary+ and ·.
- The set Z of in tegers,^2 with ordinary + and ·.
- The set Ql of rational numbers,^2 with ordinary + and ·.
- The set Z of integers,^2 with the operations: a + b = the units digit of
the ordinary sum a+ b, and a· b = 1 + the ordinary product ab. For
example, 23 + 184 = 7 and 4 · 8 = 33. - The set JR^2 = {(x,y) : x,y E JR} with (x,y) + (u,v) = (x + u,y + v) ,
the ordinary vector sum, and (x ,y) · (u,v) = xu + yv, the ordinary dot
product. [Here, JR denotes the set of real numbe r s.^2 ] - The set JR^2 with (x, y) + (u, v) = (x + u, y + v) , the ordinary vector sum,
and (x,y) · (u,v) = (xu,yv), the "pairwise" product. - The set JR^3 = {(x,y,z) : x,y,z E JR} with (x,y,z) + (u,v,w) = (x +
u, y + v, z + w), the ordinary vector sum, and (x , y, z) · (u, v, w) = (yw -
zv, - xw + zu, xv - yu), the ordinary cross product. - The set of real numbers F = {a+ b/2 : a, b E Ql}, with the ordinary
addition and multip lication of real numbers. - The set { 0, 1, 2, 3, 4} with the operations + and · defined by the tables:
+^0 1 2 3 4 0 1 2 3 4
0 0 1 2 3 4 0 0 0 0 0 0
1 1 2 3 4 0 1 0 1 2 3 4
2 2 3 4 0 1 2 0 2 4 1 3
3 3 4 0 1 2 3 0 3 1 4 2
4 4 0 1 2 3 4 0 4 3 2 1- The number systems N (natural numbers), Z (integers), IQi (rational numbers), and IR (real
numbers) will be defined formally later in this chapter. In these exercises, assume that these
number systems have the properties usually ascribed to them.