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180 Relations
(b) What’s wrong with the following proof that (S \ T ) ◦ R ⊆ (S ◦ R) \
(T ◦ R)?

Proof. Suppose (a, c) ∈ (S \ T ) ◦ R. Then we can choose some
b ∈ B such that (a, b) ∈ R and (b, c) ∈ S \ T ,so(b, c) ∈ S and
(b, c) /∈ T . Since (a, b) ∈ R and (b, c) ∈ S,(a, c) ∈ S ◦ R. Sim-
ilarly, since (a, b) ∈ R and (b, c) /∈ T ,(a, c) /∈ T ◦ R. Therefore
(a, c) ∈ (S ◦ R) \ (T ◦ R).Since(a, c)wasarbitrary,thisshowsthat
(S \ T ) ◦ R ⊆ (S ◦ R) \ (T ◦ R).

(c) Must it be true that (S \ T ) ◦ R ⊆ (S ◦ R) \ (T ◦ R)? Justify your
answer with either a proof or a counterexample.
p d12. Suppose R is a relation from A to B and S and T are relations from B
to C. Must the following statements be true? Justify your answers with
proofs or counterexamples.
(a) If S ⊆ T then S ◦ R ⊆ T ◦ R.
(b) (S ∩ T ) ◦ R ⊆ (S ◦ R) ∩ (T ◦ R).
(c) (S ∩ T ) ◦ R = (S ◦ R) ∩ (T ◦ R).
(d) (S ∪ T ) ◦ R = (S ◦ R) ∪ (T ◦ R).

4.3. More About Relations

Although we have defined relations to be sets of ordered pairs, it is sometimes
useful to be able to think about them in other ways. Often even a small change
in notation can help us see things differently. One alternative notation that
mathematicians sometimes use with relations is motivated by the fact that
in mathematics we often express a relationship between two objects x and y
by putting some symbol between them. For example, the notations x = y,
x < y, x ∈ y, and x ⊆ y express four important mathematical relationships
between x and y. Imitating these notations, if R is a relation from A to B, x ∈ A,
and y ∈ B, mathematicians sometimes write xRy to mean (x, y) ∈ R.
For example, if L is the relation defined in part 4 of Example 4.2.2, then
for any student s and dorm room r, sLr means (s,r) ∈ L, or in other words,
the student s lives in the dorm room r. Similarly, if E and T are the relations
defined in parts 5 and 6 of Example 4.2.2, then sEc means that the student s
is enrolled in the course c, and cT p means that the course c is taught by the
professor p. The definition of composition of relations could have been stated
by saying that if R is a relation from A to B and S is a relation from B to C, then
S ◦ R ={(a, c) ∈ A × C |∃b ∈ B(aRb and bSc)}.