Why Read This Book? - Index of

112 Chapter 3 Sets and Their Properties
the set of ordered pairs where the first term in the pair is an element of A and the
second is an element of B. Arelation from A to B is defined to be any subset of
A × B.
Example 3.11.1 For A ={1, 2, 3} and B ={11, 12, 13, 21, 22, 23, 31, 32, 33},
the set
is a relation from A to B. �
R ={(1, 11), (2, 21), (2, 22), (3, 31), (3, 32), (3, 33)} (3.65)
In this section, we are going to delve only into subsets of A × A, which we call
a relation on A instead of a relation from A to A.
Example 3.11.2 Let A ={1, 2, 3, 4, 5, 6}. Then
is a relation on A. �
R ={(1, 3), (1, 5), (2, 4), (2, 6), (3, 5), (4, 6)} (3.66)
You might have seen the Cartesian plane written as R × R, or more succinctly
as R 2 .IfR is a relation on the real numbers, we can represent it graphically as a
set of points in the xy-plane.
Example 3.11.3 {(x, y) : y>|x| + 1} is a relation on the real numbers
(Fig. 3.13). �
EXERCISE 3.11.4 For each of the following relations on the real numbers,
sketch the set of included points in the xy-plane.
Figure 3.13 The relation y>|x| + 1.
1
y
x