Mathematical Techniques for Computer Science (COMP11120)
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Lecture Notes on Equivalence Relations and Modular Arithmetic (COMP11120)
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Course
Mathematical Techniques for Computer Science (COMP11120)
Institution
The University Of Manchester (UOM)
Dive into the concepts of equivalence relations and modular arithmetic with these comprehensive lecture notes for COMP11120. Covering key topics such as the properties and applications of equivalence relations, and the fundamentals of modular arithmetic, these notes provide clear explanations and i...
Mathematical Techniques for Computer Science (COMP11120)
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Equivalence Relations and Modular Arithmetic
Binary relations
When consider relations from We show this directed graph
discussing relations have - as
as we seen sometimes , we a set can a as
follows
.
S to itself.
V
Instead of R
saying R from S usually say that binary
is a relation 5 to wo is a
1
relation on S
.
L
>
We can represent binary relations as a directed graph. 2 wh
-v
Y
-
X
Example : Let S = Ev ,
w
,
x
, Y, 2)
Let R be a
binary relation on S where
R ((v =
, w) , (v, x) , (2 , 2), (W , v) , (W , 2) (W x) ( -, ) (2
, , , , , 2)]
Properties of relations
Reflexivity Symmetry Transitivity
Informally a relation is
reflexive if every element
is A relation is symmetric if we can
go from one A relation is transitive if
for all s
,
sin S if there is
related to itself. element to another , then we can also go back
.
a path between two elements s and S' , then there
to s
A
binary relation R On a set S is symmetric if and only
is an
edge from s
A binary relation R is reflexive if and
only if for all
if we have for all s, s'ES if (5 5) ER ,
,
then
SES , (5 3) EIR (s' 5) R A relation R S transitive if
E
binary set is and only if
,
,
On a we
have for all s'ES if (5 5) ER s, ,
and (sis") ER
,
then 15 S") ER
,
.
We have
already seen that the identify relation for a Example : The binary relation Ron S =
Ev ,
w
,
x
, y, 2)
set S defined shown is not symmetric (v , X) ER ,
s
is as as
but (X , v) R There are other pairs
((5 5) SxSses]
.
Ig = ,
missing also
V
1
A binary relation R on S is reflexive if and
only if
"
Is &R wh Ev 2)
2 [
Example : The binary relation R on S =
,
W
,
X
, y, shown
is not transitive as (V , W) and (W , v) are in the
relation ,
but (v , v) is not.
Example : The binary relation R on S =
Ev ,
w
,
x
, y, 2) Y
shown is not reflexive as (v , v) &R
,
V
(W w),
R, and (y Y) # R
,
. 1
Symmetric Closure of a
Binary Relation
L
>
V 2 wh
The
symmetric
[
closure allows us to take a relation and
1
add a minimum number of pairs to make it
symmetric .
·
L
>
2 wh Y
Recall that the opposite relation for a
binary
relation R on a set S is defined as
Y · Rop =
((5; 3) =Sx 5/(s s) ,
ER] Transitive Closure of a
Binary Relation
The transitive closure allows us to take a relation and
add number of to make it
Reflexive
a minimum pairs
Closure of a
Binary Relation The symmetric closure of the binary relation R on the
transitive.
Set S is
given by
The reflexive closure allows us to take a relation and
add it The transitive closure of the binary relation R on the set
a minimum number of pairs to make reflexive .
RuRO = Rud(sis)eSxS((s 5) , E
R] S is
given by adding the set of pairs (Si , Sul to R
where S1 , S2 Sn ins with n = 2 such that for
The reflexive closure of R by , ...
is
given
all 1 i < n-1 we have (Si , Sin1) ER
seS]
.
Rulg =
Ru((s s) ,
If a relation R is both reflexive and symmetric ,
We can represent it as an Undirected Graph
A
i V
Z
·
su 3 2 W
y? · Y X
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