Title: Equivalence Relations and Classes
1Equivalence Relations and Classes
2Equivalence Relations
Definition
A relation R on a set A is called an equivalence
relation if it is reflexive, symmetric, and
transitive.
3Equivalence Relations
What is it?
An example
Let R be a relation on the set of people, such
that (x,y) is in R if x and y are the same age in
years.
- R is reflexive
- you are the same age as yourself
- R is symmetric
- if x is same age as y then y is same age as x
- R is transitive
- if (x,y) and (y,z) are in R then (x,z) is in R
4Equivalent Elements
Definition
Two elements related by an equivalence relation
are said to be equivalent
5Equivalence Relations
Example
Which of the following are equivalence
relations, on the set of people?
- R1 (a,b) a and b have same parents
- R2 (a,b) a and b have met
- R3 (a,b) a and b speak a common language
6Equivalence Relations
Example
- Establish if
- reflexive, i.e. (a,a) is in r
- symmetric, i.e. (a,b) and (b,a) are in r
- transitive, i.e (a,b) and (b,c) therefore (a,c)
- R1 (a,b a and b have same parents
- reflexive? symmetric? transitive?
- R2 (a,b) a and b have met
- transitive?
- R3 (a,b) a and b speak a common language
- transitive?
7Equivalence Relations
Example
Is R an equivalence relation?
8Equivalence Relations
Example
9Equivalence Relations
Example
10Equivalence Relations
Example
11Equivalence Class
Definition
Let R be an equivalence relation on the set A.
The set of all elements of A related to the
element x, also in A, are called the equivalence
class of a.
The equivalence class of a with respect to R
I.e. all elements related to a.
12Equivalence Classes
Example
What are the equivalence classes of 0 and 1 for
congruence modulo 4?
13Example
What are the equivalence classes of 0 and 1 for
congruence modulo 4?
14Equivalence Classes and Partitions
The equivalence classes of an equivalence
relation partition a set into non-empty disjoint
subsets
Let R be an equivalence relation on the set A
Proof page 411
15Example
Show that relation R consisting of pairs (x,y)
is an equivalence relation, where x and y are bit
strings and (x,y) is in R if their first 3 bits
are equal
- (x,y) is in R if
- length(x) gt 2 and length(y) gt 2
- first3bits(x) first3bits(y)
- R is reflexive
- why?
- R is symmetric
- why?
- R is transitive
- why?
- Consequently R is an equivalence relation
16Example
The pair ((a,b),(c,d)) is in R if ad bc, where
a, b, c, and d are integers. Show that R is an
equivalence relation.
- R is reflexive
- ((a,b),(a,b)) is in R because ab ba
- R is symmetric
- ((a,b),(c,d)) therefore ((c,d),(a,b))
- ad bc therefore cb da
- R is transitive
- ((a,b),(c,d)) and ((c,d),(e,f)) -gt ((a,b),(e,f))
- af be
- ad bc therefore a bc/d
- cf de therefore f de/c
- af bc/d x de/c be
- Consequently R is an equivalence relation
171. What is the equivalence class of (1,2) with
respect to the equivalence relation
(a,b)R(c,d) if ad bc 2. What does
(a,b)R(c,d) if ad bc mean?
1. (1,2)R(c,d) if 1d 2c. The equivalence class
is then the set of ordered pairs (c,d) such
that d 2c
2. R defines the set of rational numbers!
18Hey! This has got to be easier.
19Equivalence Relation
Three ways we can look at it
- A set of tuples
- A connection matrix
- A digraph
20Equivalence Relation
Reflexive
- A set of tuples
- (a,a) is in R
- A connection matrix
- diagonal is all 1s
- A digraph
- loops on nodes
21Equivalence Relation
Symmetric
- A set of tuples
- (a,b) and (b,a) are in R
- A connection matrix
- symmetric across diagonal
- A digraph
- double edges
22Equivalence Relation
Transitive
- A set of tuples
- if (a,b) and (b,c) are in R
- then (a,c) is in R
- A connection matrix
- ?
- A digraph
- triangles, ultimately a clique!
23Equivalence Relation
Example
R (a,b),(a,c),(b,a),(b,c),(c,a),(c,b),(d,e),(d,
f),(d,g), (e,d),(e,f),(e,g),(f,d),(f,e),(f,
g),(g,d),(g,e),(g,f), (a,a),(b,b),(c,c),(d,
d),(e,e),(f,f),(g,g)
24Equivalence Class
Example
d d,e,f,g e d,e,f,g f
d,e,f,g g d,e,f,g
a a,b,c b a,b,c c a,b,c
25Equivalence Relations and Classes