Equivalence Relations

1. Equivalence Relations Rosen 7.5

2. 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.

3. 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

4. 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 So, it’s the same as saying that a and b have the same remainder.

5. 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)  aMOD 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 = q1m+r – (q2m+r) =m(q1-q2) so m divides a-b.

6. 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 = b+mc. We know that b = qm + r for some r < m, so that bMODm = r . What is aMODm? a = b+mc = qm+r + mc = (q+c)m + r. So aMODm = r = bMODm

7. 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 bMOD2=aMOD2 aMOD2=bMOD2, bMOD2=cMOD2 aMOD2=cMOD2

8. Let R be the relation on the set of ordered pairs of positive integers such that ((a,b), (c,d))R iff ad=bc. 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, then ((c,d),(a,b))  R. If ((a,b),(c,d)  R, then ad = bc and cb = da since multiplication is commutative. Therefore ((c,d),(a,b))  R,

9. Let R be the relation on the set of ordered pairs of positive integers such that ((a,b), (c,d))R iff ad=bc. 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))Rand ((c,d), (e,f)) R, then ((a,b),(e,f) R. Assume that ((a,b), (c,d))Rand ((c,d), (e,f)) R. Then ad = cb and cf = de. This implies that a/b = c/d and that c/d = e/f, so a/b = e/f which means that af = be. Therefore ((a,b),(e,f)) R. (remember we are using positive integers.)

10. 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 0*m, so a-a is divisible by m.

11. 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).

12. 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) = jm+km = (j+k)m Since j+k is an integer, then m divides (a-c) so ac(MOD m).

13. 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 [a]R. I.e., [a]R = {s | (a,s)  R} Note that an equivalence class is a subset of A created by R. If b  [a]R, b is called a representative of this equivalence class.

14. Example Let A be the set of all positive integers and let R = {(a,b) | a MOD 3 = b MOD 3} How many distinct equivalence classes (rank) does R create? 3

15. 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

16. 5 4 2 1 8 6 7 3 Let A = {1,2,3,4,5,6,7,8} and let R = {(a,b)|ab(MOD 3)} be a relation on A.

17. Partition • Let S1, S2, …, Sn be a collection of subsets of A. Then the collection forms a partitionof 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.

18. 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

19. How many equivalence relations can there be on a set A with n elements? A has three elements Rank = 3 Rank = 2 Rank = 1

20. How many equivalence relations can there be on a set A with n elements? A has four elements Rank = 4 Rank = 1

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 = 2

23. How many equivalence relations can there be on a set A with n elements? A has four elements Rank = 3

24. 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 • 15 for n = 4 • ? for n = 5 • Is there recurrence relation or a closed form solution?