Bridge to Abstract Mathematics: Mathematical Proof and Structures

22 SETS Chapter 1

DEFINITION 7

Given sets A and B, we define the cartesian product A x B (read "A cross B,"

cartesian product is often called cross product) by the rule A x B =

{(a, b) 1 a E A, b E 6).

Thus A x B consists of all possible distinct ordered pairs whose first ele-

ments come from A and whose second elements come from B. An object x

is an element of A x B if and only if there exist a E A and b E B such that

x = (a, b). Note that there is nothing in the definition to prevent A and B

from being the same set.

EXAMPLE 9 Given A = ( 1,2,3) and B = (w, x, y, z), describe A x B by

the roster method.

Solution A x B = ((44, (1, x), (1, y), (1,4, (294, (2, x), (2, Y), (2,z), (394,

(3, x), (3, y), (3,~)). We have chosen to list these ordered pairs by first

pairing the number 1 with each letter, then 2, then 3. We could have used

some other approach, such as pairing each of the three numbers in A with

w, then with x, y, and z, respectively. Any such approach is all right as

long as all the ordered pairs are accounted for, since the order in which

the ordered pairs in A x B are listed is, of course, inconsequential. Note

that (y, 2) is not an element of A x B. You should be able to name a set

closely related to A x B that contains the ordered pair (y, 2), as well as

list the elements of A x A and B x B. 0

EXAMPLE 10 Describe A x B if A = B = R. Describe geometrically the

subset I x J of R x R, where I = [3,7] and J = (-2,2).

Solution R x R is the set of all ordered pairs of real numbers, usually pic-

tured by a two-dimensional graph, as shown in Figure l.la, with points

in the plane corresponding to ordered pairs. These in turn are labeled

Figure 1.1 (a) The x-y coordinate system; (b) [3,7] x ( - 2,2).

Y Y