cse 1400 applied discrete mathematics
relations
2
Problems on Relations
18
Abstract
A relation
∼
describes how
things
are connected. That a
thing
a
is
related to a
thing
b
can be represented by
1
.
An
ordered pair
(
a
,
b
)
.
2
.
An directed edge
a
•
•
b
.
3
.
Or more commonly, simply using relational notation
a
∼
b
.
A relation is
1
.
A set
G
of
ordered pairs
.
2
.
A directed graph
G
of nodes and edges.
3
.
A matrix of
True
and
False
values.
Higher-dimensional relations among
a
,
b
,
c
or more parameters can
be defined. Higher-dimensional relations occur as tables in relational
databases and as data in multi-variable problems.
Relations and Their Graphs
A
relation is a set of ordered pairs
.
A relation can be pictured of as a graph
G
.
•
a
•
b
c
•
d
•
•
e
•
f
h
•
•
g
This
graph represents the relation
G
=
{
(
a
,
b
)
,
(
b
,
c
)
,
(
c
,
d
)
,
(
d
,
h
)
,
(
h
,
f
)
,
(
f
,
e
)
,
(
e
,
a
)
}
G
=
{
(
x
,
y
)
:
x
∼
y
}
=
{
(
x
,
y
)
:
x
is related to
y
.
}
There are many examples of relations. You are, no doubt, familiar
with relations among people: Mother-Daughter, Father-Son, Parent-
Child, Aunt-Nephew. Familial relations often become murky. We
study relations that can be precisely defined. A few common rela-
tions are equality, congruence mod
n
, less than, divides, subset, and
perpendicular.
In this course, relationships will be between two
things
a
and
b
.
Relationships among 3 or more things are common and useful, but
The course studies
binary
relations.
these ideas are not within the scope of this course.
A Relation’s Domain, Co-domain, and Range
T
he things
involved in a relation need
names
: C
all them
the things
x
and
y
. Write
x
∼
y
to express the phrase
“
x
is
related
to
y
.”