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