Introduction
Orders and Equivalences
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Orders and Equivalences
15-2
Previous Lecture
Properties of binary relations
Equivalence relations
s
reflexivity
s
symmetricity
s
transitivity
s
anti-symmetricity
Discrete Mathematics – Orders and Equivalences
15-3
Properties of binary relations
Reflexivity
A binary relation
R
⊆
A
×
A
is said to be
reflexive
if
(a,a)
∈
R
for all
a
∈
A.
Symmetricity
A binary relation
R
⊆
A
×
A
is said to be
symmetric
if, for
any
a,b
∈
A,
if
(a,b)
∈
R
then
(b,a)
∈
R.
Transitivity
A binary relation
R
⊆
A
×
A
is said to be
transitive
if, for
any
a,b,c
∈
A,
if
(a,b)
∈
R
and
(b,c)
∈
R
then
(a,c)
∈
R.
Anti-symmetricity
A binary relation
R
⊆
A
×
A
is said to be
anti-symmetric
if,
for any
a,b
∈
A,
if
(a,b)
∈
R
and
(b,a)
∈
R
then
a =b.
Discrete Mathematics – Orders and Equivalences
15-4
Equivalence relations
A binary relation
R
on a set
A
is said to be an
equivalence
relations
if it is reflexive, symmetric, and transitive.
Let
R
⊆
People
×
People.
Pair
(a,b)
∈
R
if and only if
a
and
b
are of the same age.
Equivalence classes.
Take
a
∈
A.
The set
C(a) = { b | (a,b)
∈
R}
is called the
equivalence class
of
a.
For example,