Part A. Relations.
Definition.
A
relation R
defined over sets
A
and
B
is a (i.e. any) subset of
A
×
B
, that is,
R
⊂
A
×
B
.
Such
a relation is a
binary
relation; the
degree
of
R
is 2.
Definition.
An
n
ary relation
R
over sets
A
1
,
A
2
,
…, A
n
, n
≥
2, is a subset of the cartesian product
The
degree
of
R
is
n
.
Definition.
Let
R
⊂
A
×
B
and
S
⊂
B
×
C
denote two relations.
The
composition
of
R
and
S
, denoted
R
S
, is a binary relation over
A
and
C
defined as follows:
R
S
=
{(
a
,
c
)
a
∈
A
and
c
∈
C
, and there exists
b
∈
B
such that
aRb
and
bSc
}.
Theorem.
Let
R
⊂
A
×
B
,
S
⊂
B
×
C
, and
T
⊂
C
×
D
denote three binary relations.
Then relation
compositions satisfy the following associative law:
(
R
S
)
T
=
R
(
S
T
).
Theorem.
Let
R
⊂
A
×
B
,
S
⊂
B
×
C
, and
T
⊂
B
×
C
denote 3 binary relations.
Then
(1)
R
(
S
∪
T
) = (
R
S
)
∪
(
R
T
);
(2)
R
(
S
∩
T
)
⊂
(
R
S
)
∩
(
R
T
).
(In general, the two sides of (2) are not equal.)
Definition.
Let
R
⊂
A
×
A
be a binary relation.
(1)
R
is
reflexive
if for all
a
∈
A
,
aRa
, i,e., (
a
,
a
)
∈
R
.
(In words, each element
a
of
A
is related to itself via
R
.)
(2)
R
is
irreflexive
if for all
a
∈
A
, (
a
,
a
)
∉
R
.
(In words, each element
a
of
A
is not related to itself via
R
.)
(3)
R
is
symmetric
if
aRb
implies
bRa
, i.e.,
for all (
a
,
b
)
∈
R
, (
b
,
a
)
∈
R
.
(In words, whenever
a
is related
to
b
,
b
is related to
a
.)
(4)
R
is
anti

symmetric
if
aRb
and
bRa
imply
a
=
b
.
Equivalently, this means if
a
≠
b
, then (
a
,
b
)
∈
R
implies (
b
,
a
)
∉
R
. (In words, whenever
a
is related to
b
, and if
a
≠
b
, then b is not related to
a
.)
(5)
R
is
transitive
if
aRb
and
bRc
imply
aRc
, i.e., if (
a
,
b
)
∈
R
and (
b
,
c
)
∈
R
, then
(
a
,
c
)
∈
R
.
(In words,
whenever
a
is related to
b
, and
b
is related to
c
, then
a
is related to
c
.)
Definition.
Consider a binary relation
R
⊂
A
×
B
.
The
inverse
of
R
, denoted
R
–1
, is a binary relation
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Binary relation, Bijection, Inverse relation

Click to edit the document details