7.1 RELATIONS 231EXAMPLE 2 Define a relation R5 on R by R5 = {(x, y) E R x R 13 < x < 8,
-2 I y < 4). This relation may be pictured as a rectangle in the xy
plane, open on the left and top, closed on the right and bottom, as pic-
tured in Figure 7.1. You should draw a pictorial representation of the
relation on R, ((x, y) 1(x2/9) + (y2/49) t 1).
EXAMPLE 3 Define reladions R,, R,, and R, on the set H of all human
beings living in the year 1987 by R, = {(x, y)lx is older than y or the
same age as y}, R, = ((x, y)l y is a biological parent of x}, and R, =
{(x, y)lx and y are both male or both female).
EXAMPLE 4 Define relations R,, R,,, and R, , on Z by the rules R, =
((m, n)lm and n are both even or both odd}, R,, = ((m, n)lm and n are
both nonnegative}, and R, , = {(m, n) 1 m is less than or equal to n). 0
EXAMPLE 5 Define relations R,,, R13, and R14 on Z by the rules R,, =
{(m, n)lm divides n) (recall the paragraph preceding Example 7, Article
5.4), R13 = ((m, n)15 divides the difference m - n of m and n), and
R,, = ((m, n) ( m equals n). 0
EXAMPLE 6 Let X be any finite set and let A = P(X). Define relations
R,,, R16, and R,, on A by R,, = ((M, N)IM is a subset of N), R16 =
((M, N)I M and N are disjoint), and R,, = ((M, N)I M and N have the
same number of elements). 0
A common way of viewing a relation is from a dynamic, rather than a
static, point of view. Frequently, we think of a relation not primarili as a
set of ordered pairs, but rather, as a relationship, where the relationship
exists between precisely those pairs of objects that occur together in an
Y Figure 7.1 Graphic representation of the
relation R, = {(x, y)13 < x I 8,
-2 < y < 4).
r-----
I
I
I
I
IX