Introduction
Equivalences
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Equivalences
142
Previous Lecture
Cartesian products of two and more sets
Cardinality and other properties of Cartesian products
Binary, ternary and
kary relations
Describing binary relations
Discrete Mathematics – Equivalences
143
Properties of Binary Relations – Reflexivity
From now on we consider only binary relations from a set
A
to the
same set
A.
That is such relations are subsets of
A
×
A.
A binary relation
R
⊆
A
×
A
is said to be
reflexive
if
(a,a)
∈
R
for all
a
∈
A.
(a,b)
∈
R
⊆
Z
×
Z
if and only if
a
≤
b
This relation is reflexive, because
a
≤
a
for all
a
∈
Z
Matrix:
1
*
*
*
*
1
*
*
*
*
1
*
*
*
*
1
1’s
on the diagonal
Graph:
Loops at every vertex
Discrete Mathematics – Equivalences
144
Properties of Binary Relations – 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.
The relation
Brotherhood
(`x
is a brother of y’) on the set of men is
symmetric, because if
a
is a brother of
b
then b
is a brother of
a
Matrix:
1
1
1
1
Matrix is symmetric w.r.t.
the diagonal
Graph:
Graph is symmetric
Discrete Mathematics – Equivalences
145
Properties of Binary Relations – Transitivity
A binary relation
R
⊆
A
×
A
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 Spring '10
 AndreiBulatov
 Equivalence relation, Transitive relation, equivalence classes, equivalences

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