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More About Relations 183
R. The relation E −1 ◦ E from Example 4.2.4 was a relation on the set S, and
E ◦ E −1 was a relation on C.

Example 4.3.1. Here are some more examples of relations on sets.

1. Let A ={1, 2} and B = P (A) = {∅, {1}, {2}, {1, 2}} as in part 3 of Exam-
ple 4.2.2. Let S ={(x, y) ∈ B × B | x ⊆ y}= {(∅,∅), (∅, {1}), (∅, {2}),
(∅, {1, 2}), ({1}, {1}), ({1}, {1, 2}), ({2}, {2}), ({2}, {1, 2}), ({1, 2}, {1, 2})}.
Then S is a relation on B.
2. Suppose A is a set. Let i A ={(x, y) ∈ A × A | x = y}. Then i A is a relation
on A. (It is sometimes called the identity relation on A.) For example, if
A ={1, 2, 3}, then i A ={(1, 1), (2, 2), (3, 3)}. Note that i A could also be
defined by writing i A ={(x, x) | x ∈ A}.
3. For each positive real number r, let D r ={(x, y) ∈ R × R | x and y differ
by less than r, or in other words |x − y| < r}. Then D r is a relation on R.

Suppose R is a relation on a set A. If we used the method described earlier
to draw a picture of R, then we would have to draw two copies of the set A
and then draw edges from one copy of A to the other to represent the ordered
pairs in R. An easier way to draw the picture would be to draw just one copy
of A and then connect the vertices representing the elements of A with edges
to represent the ordered pairs in R. For example, Figure 3 shows a picture of
the relation S from part 1 of Example 4.3.1. Pictures like the one in Figure 3
are called directed graphs.

Figure 3