Equivalence Relations and Classes

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

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

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

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}

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}

Equivalence Relations Example Is R an equivalence relation?

Equivalence Relations Example

Equivalence Relations Example

Equivalence Relations Example

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.

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

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

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

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

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

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!

Hey! This has got to be easier.

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

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

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

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

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)}

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}

Equivalence Relations and Classes