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Equivalence Relations Rosen 7.5. Equivalence Relation A relation on a set A is called an equivalence relation if it is –Reflexive –Symmetric –Transitive.

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Presentation on theme: "Equivalence Relations Rosen 7.5. Equivalence Relation A relation on a set A is called an equivalence relation if it is –Reflexive –Symmetric –Transitive."— Presentation transcript:

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  q 1,q 2,r  Z such that a = q 1 m + r and b = q 2 m + r. a-b = q 1 m+r – (q 2 m+r) =m(q 1 -q 2 ) 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))  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 = 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 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 partition of A if the subsets are nonempty, disjoint and exhaust A. S i   S i  S j =  if i  j  S i = 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 one element. One equivalence class, rank =1. A has two elements rank = 2 rank = 1

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


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