1. Equivalence Relations and Classes

2. Equivalence Relations
Definition
A relation R on a set A is called an equivalence relation
if it is reflexive, symmetric, and transitive.

3. Equivalence 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

4. Equivalent Elements
Definition
Two elements related by an equivalence relation
are said to be equivalent

5. Equivalence 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}

6. Equivalence 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}

7. Equivalence Relations
Example
Is R an equivalence relation?

8. Equivalence Relations
Example

9. Equivalence Relations
Example

10. Equivalence Relations
Example

11. Let R be an equivalence relation on the set A. The set of
Equivalence 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.

12. Equivalence Classes
Example
What are the equivalence classes of 0 and 1
for congruence modulo 4?

13. Example What are the equivalence classes of 0 and 1
for congruence modulo 4?

14. Equivalence 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

15. Example
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) > 2 and length(y) > 2
first3bits(x) = first3bits(y)
R is reflexive
why?
R is symmetric
R is transitive
Consequently R is an equivalence relation

16. Example
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)) -> ((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

17. 1. 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!

18. Hey! This has got to be easier.

19. Equivalence Relation
Three ways we can look at it
A set of tuples
A connection matrix
A digraph

20. Equivalence Relation
Reflexive
A set of tuples
(a,a) is in R
A connection matrix
diagonal is all 1’s
A digraph
loops on nodes a b

21. Equivalence Relation
Symmetric
A set of tuples
(a,b) and (b,a) are in R
A connection matrix
symmetric across diagonal
A digraph
double edges a b

22. Equivalence 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! a b c a b c

23. Equivalence Relation
Example a b c d g e f
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)}

24. Equivalence Class
Example a b c d g e f
[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}

25. Equivalence Relations and Classes