McCombs Math 381
Relations on Sets
1
Basic Idea:
Given a set
A
, a
relation
R
defined on set
A
is a rule that describes how to
compare the elements of set
A
to one another. In some cases, this rule will be
described explicitly by some formula or equation. In other cases, this rule will
be described by listing ordered pairs of set elements.
Formal Definition:
Given a set
A
, a
relation
R
on set
A
is a set of ordered pairs comprised
of the elements of set
A
.
We use the notation:
R
=
x
,
y
( )
:
x
!
A
,
y
!
A
{ }
.
Important Notation:
xRy
corresponds to the ordered pair
x
,
y
( )
Important Fact:
Every relation
R
:
A
!
A
is a subset of the set
A
!
A
, i.e.
R
!
A
"
A
.
Relations Between Sets:
Given sets
A
and
B
, a
relation
R
from
A
to
B
is a set of ordered pairs
comprised of the elements of the sets
A
and
B
.
We use the notation:
R
=
x
,
y
( )
:
x
!
A
,
y
!
B
{ }
Special Properties:
Suppose
R
is a relation on set
A
.
1.
R
is
reflexive
means that for every element
x
!
A
we have
xRx
.
In other words, for every element
x
!
A
, the ordered pair
x
,
x
( )
!
R
.
2.
R
is
symmetric
means that for every element
x
,
y
!
A
we have
xRy
!
yRx
.
In other words, if
x
,
y
( )
!
R
, then
y
,
x
( )
!
R
.
3.
R
is
transitive
means that for every element
x
,
y
,
z
!
A
,
we have
xRy
!
yRz
( )
"
xRz
.
In other words, if
x
,
y
( )
!
R
and
y
,
z
( )
!
R
, then
x
,
z
( )
!
R
.
4.
R
is
irreflexive
means that for every element
x
!
A
we have
x
.
In other words, for every element
x
!
A
, the ordered pair
x
,
x
( )
!
R
.