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Chapter 5
Section 5.1
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Section Summary
Relations and Functions
Properties of Relations
Reflexive Relations
Symmetric and Antisymmetric Relations
Transitive Relations
Combining Relations
Relations
Definition:
A
relation R
from a set
A
to a set
B
is a
subset
R
⊆
A
×
B.
Example
:
Let
A =
{
0
,
1,2
} and
B =
{
a,b
}
{(
0,
a
)
,
(
0,
b
)
,
(
1,
a
)
,
(
2,
b
)} is a relation from
A
to
B
.
We can represent relations from a set
A
to a set
B
graphically or using a table:
Relations are more general than
functions. A function is a relation
where exactly one element of
B
is
related to each element of
A.
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Relation on a Set
Definition:
A relation
R on a set S
is a subset of
S
×
S
or a relation from
S
to
S
.
Example
:
Suppose that
S =
{
a,b,c
}. Then
R =
{(
a,a
)
,
(
a,b
)
,
(
a,c
)} is
a relation on
S
.
Let
S =
{
1, 2, 3, 4
}. The ordered pairs in the relation
R = {(
a
,
b
) 
a
divides
b
} are
(1,1), (1, 2), (1,3), (1, 4), (2, 2), (2, 4), (3, 3), and
(4, 4).
Relation on a Set (
cont.
)
Question
: How many relations are there on a set
A
?
Solution
:
Because a relation on
A
is the same thing as a
subset of
A
⨉
A
, we count the subsets of
A
×
A
. Since
A
×
A
has
n
2
elements when
A
has
n
elements, and a set
with
m
elements has
2
m
subsets, there are
subsets of
A
×
A
. Therefore,
there are
relations on a set
A
.
2


2
A
2


2
A
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4
Relations on a Set (
cont
.)
Example
: Consider these relations on the set of integers:
R
1
= {(
a
,
b
) 
a
≤
b
},
R
4
= {(
a
,
b
) 
a
=
b
},
R
2
= {(
a
,
b
) 
a
>
b
},
R
5
= {(
a
,
b
) 
a
=
b
+ 1},
R
3
= {(
a
,
b
) 
a
=
b
or
a
=
−b
},
R
6
= {(
a
,
b
) 
a
+
b
≤ 3}.
Which of these relations contain each of the pairs
(1,1), (1, 2), (2, 1), (1, −1), and (2, 2)?
Solution
: Checking the conditions that define each relation, we see
that the pair
(1,1) is in
R
1
,
R
3
,
R
4
, and
R
6
: (1,2) is in
R
1
and
R
6
: (2,1) is in
R
2
,
R
5
, and
R
6
: (1, −1) is in
R
2
,
R
3
, and
R
6
: (2,2) is in
R
1
,
R
3
, and
R
4
.
Note that these relations are on an infinite set and each of these relations is an
infinite set.
Properties of Relations
Let R be a relation defined on a nonempty set S. Then
R is:
Reflexive
𝑖? ??? ??? ?𝑙𝑙 ? ∈ ?, ?ℎ??
?, ?
∈ ?
Symmetric
𝑖??ℎ?????? ???, ?ℎ?? ??? ??? ?𝑙𝑙 ?, ? ∈ ?,
?? 𝑖? ??ℎ?? ????? 𝑖?
?, ?
∈ ?, ?ℎ??
?, ?
∈ ?
Transitive
𝑖? ?ℎ?????? ??? ??? ??? ?ℎ?? ??? ??? ?𝑙𝑙 ?, ?, ? ∈ ?
?? 𝑖? ??ℎ?? ????? 𝑖?
?, ?
∈ ? ???
?, ?
∈ ?, ?ℎ?? (?, ?) ∈ ?