Equivalence RelationsA relation R on a set A is called an equivalence relation if it satisfies following three properties:
Example: Let A = {1, 2, 3, 4} and R = {(1, 1), (1, 3), (2, 2), (2, 4), (3, 1), (3, 3), (4, 2), (4, 4)}. Show that R is an Equivalence Relation. Solution: Reflexive: Relation R is reflexive as (1, 1), (2, 2), (3, 3) and (4, 4) ∈ R. Symmetric: Relation R is symmetric because whenever (a, b) ∈ R, (b, a) also belongs to R. Example: (2, 4) ∈ R ⟹ (4, 2) ∈ R. Transitive: Relation R is transitive because whenever (a, b) and (b, c) belongs to R, (a, c) also belongs to R. Example: (3, 1) ∈ R and (1, 3) ∈ R ⟹ (3, 3) ∈ R. So, as R is reflexive, symmetric and transitive, hence, R is an Equivalence Relation. Note1: If R_{1}and R_{2} are equivalence relation then R_{1}∩ R_{2} is also an equivalence relation.Example: A = {1, 2, 3} Note2: If R_{1}and R_{2} are equivalence relation then R_{1}∪ R_{2} may or may not be an equivalence relation.Example: A = {1, 2, 3} Hence, Reflexive or Symmetric are Equivalence Relation but transitive may or may not be an equivalence relation. Inverse RelationLet R be any relation from set A to set B. The inverse of R denoted by R^{1} is the relations from B to A which consist of those ordered pairs which when reversed belong to R that is: R^{1} = {(b, a): (a, b) ∈ R} Example1: A = {1, 2, 3} Solution: R = {(1, y), (1, z), (3, y) Note1: Domain and Range of R^{1} is equal to range and domain of R.Example2: R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (3, 2)} Note2: If R is an Equivalence Relation then R^{1} is always an Equivalence Relation.Example: Let A = {1, 2, 3} Note3: If R is a Symmetric Relation then R^{1}=R and viceversa.Example: Let A = {1, 2, 3} Note4: Reverse Order of Law
