Part A. Equivalence Relations.
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.
A binary relation
R
⊂
A
×
A
is an
equivalence
relation if
R
is reflexive, symmetric, and
transitive.
Definition.
Let
m
≥
2 be an integer.
Define a binary relation called
modulus m
and denoted
≡
m
, over the
set of integers
Z
, as follows:
for integers
a
and
b
,
a
≡
m
b
iff
m
 (
a
–
b
), that is,
a
–
b = mq
for some integer
q
.
Two numbers
a
and
b
such that
a
≡
m
b
, are called
congruent mod m
, often denoted
a
≡
b
(mod
m
).
Theorem.