Equivalence Relations

1
Equivalence Relations

• Rosen 6.5

2
Some preliminaries

• Let a be an integer and m be a positive integer.
  We denote by a MOD m the remainder when a is
  divided by m.
• If r a MOD m, then
• a qm r and 0? r ? m, q?Z
• Examples
• Let a 12 and m 5, 12MOD5 2
• Let a -12 and m 5, 12MOD5 3

3
a?b(MOD m)

• If a and b are integers and m is a positive
  integer, then a is congruent to b modulo m if m
  divides a-b.
• (a-b)MODm 0
• (a-b) qm for some q?Z
• Notation is a?b(MOD m)
• aMODm bMODm iff a?b(MOD m)
• 12MOD5 17MOD5 2
• (12-17)MOD 5 -5MOD5 0

4
Prove that a MOD m b MOD m iff a?b(MOD m)

• Proof We must show that
• aMODm bMODm ? a?b(MOD m)
• and that a?b(MOD m) ? MOD m b MOD m
• First we will show that aMODm bMODm ? a?b(MOD
  m)
• Suppose aMODm bMODm, then ? q1,q2,r?Z such that
  a q1m r and b q2m r.
• a-b q1mr (q2mr) m(q1-q2) so m divides a-b.

5
Prove that a MOD m b MOD m iff a?b(MOD m)

• Next we will show that a?b(MOD m) ? aMOD m
  bMODm.
• Assume that a?b(MOD m) . This means that m
  divides a-b, so a-b mc for c?Z.
• Therefore a bmc. We know that b qm r for
  some r lt m, so that bMODm r .
• What is aMODm?
• a bmc qmr mc (qc)m r. So aMODm r
  bMODm

6
Equivalence Relation

• A relation on a set A is called an equivalence
  relation if it is
• Reflexive
• Symmetric
• Transitive
• Two elements that are related by an equivalence
  relation are called equivalent.
• Example A 2,3,4,5,6,7 and R (a,b) a MOD
  2 b MOD 2

aMOD2 aMOD2 aMOD2 bMOD2 ?bMOD2aMOD2 aMOD2bMO
D2, bMOD2cMOD2 ?aMOD2cMOD2
7
Let R be the relation on the set of ordered pairs
of positive integers such that ((a,b), (c,d))?R
iff adbc. Is R an equivalence relation?

• Proof We must show that R is reflexive,
  symmetric and transitive.
• Reflexive We must show that ((a,b),(a,b)) ? R
  for all pairs of positive integers. Clearly ab
  ab for all positive integers.
• Symmetric We must show that ((a,b),(c,d) ? R,
  the ((c,d),(a,b)) ? R. If ((a,b),(c,d) ? R, then
  ad bc -gt cb da since multiplication is
  commutative. Therefore ((c,d),(a,b)) ? R,

8
Let R be the relation on the set of ordered pairs
of positive integers such that ((a,b), (c,d))?R
iff adbc. Is R an equivalence relation?

• Proof We must show that R is reflexive,
  symmetric and transitive.
• Transitive We must show that if ((a,b), (c,d))?R
  and ((c,d), (e,f)) ?R, then ((a,b),(e,f) ?R.
  Assume that ((a,b), (c,d))?R and ((c,d), (e,f))
  ?R. Then ad cb and cf ed. This implies that
  a/b c/d and that c/d e/f, so a/b e/f which
  means that af eb. Therefore ((a,b),(e,f) ?R.
  (remember we are using positive integers.)

9
Prove that R a?b(MOD m) is an equivalence
relation on the set of integers.

• Proof We must show that R is reflexive,
  symmetric and transitive. (Remember that a?b(MOD
  m) means that (a-b) is divisible by m.
• First we will show that R is reflexive.
• a-a 0 and 0m, so a-a is divisible by m.

10
Prove that R a?b(MOD m) is an equivalence
relation on the set of integers.

• We will show that R is symmetric. Assume that
  a?b(MOD m). Then (a-b) is divisible by m so
  (a-b) qm for some integer q. -(a-b) (b-a)
  -qm. Therefore b?a(MOD m).

11
Prove that R a?b(MOD m) is an equivalence
relation on the set of integers.

• We will show that R is transitive. Assume that
  a?b(MOD m) and that b?c(MOD m). Then ? integers
  j,k such that (a-b) jm, and (b-c) km.
• (a-b)(b-c) (a-c) jmkm (jk)m
• Since jk is an integer, then m divides (a-c) so
  a?c(MOD m).

12
Equivalence Class

• Let R be an equivalence relation on a set A. The
  set of all elements that are related to an
  element of A is called the equivalence class of
  a.
• The equivalence class of a with respect to R is
  denoted aR. I.e., aR s (a,s) ? R
• Note that an equivalence class is a subset of A
  created by R.
• If b ? aR, b is called a representative of this
  equivalence class.

13
Example

• Let A be the set of all positive integers and let
  R (a,b) a MOD 3 b MOD 3
• How many equivalence classes (rank) does R
  create?
• 3

14
Digraph Representation

• It is easy to recognize equivalence relations
  using
• digraphs.
• The subset of all elements related to a
  particular element forms a universal relation
  (contains all possible arcs) on that subset.
• The (sub)digraph representing the subset is
  called a complete (sub)digraph. All arcs are
  present.
• The number of such subsets is called the rank of
  the equivalence relation

15
Let A 1,2,3,4,5,6,7,8 and let R
(a,b)a?b(MOD 3) be a relation on A.
16
Partition

• Let S1, S2, , Sn be a collection of subsets of
  A. Then the collection forms a partition of A if
  the subsets are nonempty, disjoint and exhaust A.
• Si ? ?
• Si?Sj ? if i ? j
• ?Si A
• If R is an equivalence relation on a set S, then
  the equivalence classes of R form a partition of
  S.

17
How many equivalence relations can there be on a
set A with n elements?
A has two elements
A has one element. One equivalence class, rank
1.
rank 2
rank 1
18
How many equivalence relations can there be on a
set A with n elements?
A has three elements
Rank 3
Rank 2
Rank 1
19
How many equivalence relations can there be on a
set A with n elements?
A has four elements
Rank 4
Rank 1
20
How many equivalence relations can there be on a
set A with n elements?
A has four elements
Rank 2
21
How many equivalence relations can there be on a
set A with n elements?
A has four elements
Rank 2
22
How many equivalence relations can there be on a
set A with n elements?
A has four elements
Rank 3
23
How many equivalence relations can there be on a
set A with n elements?

• 1 for n 1
• 2 for n 2
• 5 for n 3
• 13 for n 4
• ? for n 5
• Is there recurrence relation or a closed form
  solution?