4.5 Equivalence Relations

A relation R on a set A is called an equivalence relation if it is reflexive, symmetric and transitive. Reflexive: has one node cycles at each vertex. Relation Example: {(1,1), (2,2)} Symmetric: if you have (a,b) you also have (b,a). Relation Example: {(1,2), (2,1)} Transitive: if you have (a,b) and (b,c), you must have (a,c). Relation Example: {(1,2), (2,3), (1,3)}

Equivalence Relation Example: Verify R is an equivalence relation: -Reflexive: (1,1),(2,2),(3,3),(4,3) -Symmetric: (1,2),(2,1),(3,4),(4,3) -Transitive: Does not break the transitive rule. We only have a (a,b). We don’t have a (b,c) and therefore we are not required to have an (a,c).

Equivalence Relations and Partitions Find the equivalence relation determined by P First look at the block: {1,2,3} Reflexive: (1,1), (2,2), (3,3) Symmetric and transitive: a,b a,c b,c (1,2), (2,1),(1,3),(3,1),(2,3),(3,2) Now look at block {4}, Reflexive: (4,4) Result: R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1),(1,3),(3,1),(2,3),(3,2)}

A partition set A determined by the equivalence relation R on A is denoted A/R. A/R is a collection of equivalence classes – sets that contain elements that are related to each other.

Equivalence relation is reflexive, symmetric and transitive. R = {(a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(b,e),(e,b),(c,d),(c,f),(d,c), (d,f),(f,c),(f,d)} The partition or the equivalence relation R on A: A/R = {{a},{b,e},{c,d,f}} 1. a is only related to iself 2. b is related to e and e is related to b 3. c is related to d d is related to c c is related to f f is related to c d is related to f f is related to d In other words, c,d and f are all related to each other.