⊂ R . Hence, R = R . Example Let f : X → Y be a function. Define a relation R in X given by R = {( a , b ): f ( a ) = f ( b )}.
Examine whether R is an equivalence relation or not. Solution For every a ∈ X, ( a , a ) ∈ R, since f ( a ) = f ( a ), showing that R is reflexive. Similarly, ( a , b ) ∈ R ⇒ f ( a ) = f ( b ) ⇒ f ( b ) = f ( a ) ⇒ ( b , a ) ∈ R. Therefore, R is symmetric.
Further, ( a , b ) ∈ R and ( b , c ) ∈ R ⇒ f ( a ) = f ( b ) and f ( b ) = f ( c ) ⇒ f ( a ) = f ( c ) ⇒ ( a , c ) ∈ R, which implies that R is transitive. Hence, R is an equivalence relation. Example Find the number of all one-one functions from set A = { , , } to itself. Solution One-one function from { , , } to itself is simply a permutation on three symbols , , .
Therefore, total number of one-one maps from { , , } to itself is same as total number of permutations on three symbols , , which is ! = . Example Let A = { , , }. Then show that the number of relations containing ( , ) and ( , ) which are reflexive and transitive but not symmetric is three.
Solution The smallest relation R containing ( , ) and ( , ) which is reflexive and transitive but not symmetric is {( , ), ( , ), ( , ), ( , ), ( , ), ( , )}. Now, if we add the pair ( , ) to R to get R , then the relation R will be reflexive, transitive but not symmetric. Similarly,