Page 186 - HOW TO PROVE IT: A Structured Approach, Second Edition

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172 Relations
Definition 4.2.1 does not require that a set of ordered pairs be defined as the
truth set of some statement for the set to be a relation. Although thinking about
truth sets was the motivation for this definition, the definition says nothing
explicitly about truth sets. According to the definition, any subset of A × B is
to be called a relation from A to B.

Example 4.2.2. Here are some examples of relations from one set to another.

1. Let A ={1, 2, 3}, B ={3, 4, 5}, and R ={(1, 3), (1, 5), (3, 3)}. Then R ⊆
A × B,so R is a relation from A to B.
2. Let G ={(x, y) ∈ R × R | x > y}. Then G is a relation from R to R.
3. Let A ={1, 2} and B = P (A) = {∅, {1}, {2}, {1, 2}}. Let E ={(x, y) ∈
A × B | x ∈ y}. Then E is a relation from A to B. In this case, E =
{(1, {1}), (1, {1, 2}), (2, {2}), (2, {1, 2})}.
For the next three examples, let S be the set of all students at your school,
R the set of all dorm rooms, P the set of all professors, and C the set of all
courses.
4. Let L ={(s,r) ∈ S × R | the student s lives in the dorm room r}. Then L
is a relation from S to R.
5. Let E ={(s, c) ∈ S × C | the student s is enrolled in the course c}. Then E
is a relation from S to C.
6. Let T ={(c, p) ∈ C × P | the course c is taught by the professor p}. Then
T is a relation from C to P.

So far we have concentrated mostly on developing your proof-writing skills.
Another important skill in mathematics is the ability to understand and apply
new definitions. Here are the definitions for several new concepts involving
relations. We’ll soon give examples illustrating these concepts, but first see if
you can understand the concepts based on their definitions.

Definition 4.2.3. Suppose R is a relation from A to B. Then the domain of R
is the set
Dom(R) ={a ∈ A |∃b ∈ B((a, b) ∈ R)}.

The range of R is the set

Ran(R) ={b ∈ B |∃a ∈ A((a, b) ∈ R)}.

The inverse of R is the relation R −1 from B to A defined as follows:
R −1 ={(b, a) ∈ B × A | (a, b) ∈ R}.