RELATION


☞  Cartesian product of sets

Let A and B be non-empty sets. The Cartesian product of A and B denoted by A×B, is the set defined by A×B= {(a,b): aϵA, bϵB}

A×B is the set of all ordered pairs (a,b) the first element of the pair being an element of A and the second being an element of B. 

Example1: Let A= {a,b,c}, B={d,e}  

                         A×B= {(a,d), (a,e), (b,d), (b,e), (c,d), (c,e)}

☞  RELATION

Let A and B be two non-empty sets. Then a relation ρ from A to B is a subset of A×B.

Example2: ρ= {(a,d), (c,e)} ⊆ A×B (where A×B is given in example1)

So ρ is a relation from A to B.

☞ Domain of a Relation

For any relation ρ from A to B the domain is the set of all those elements a ϵ A s.t. (a,b) ϵ ρ for some b ϵ B.

Dom(ρ)= {a ϵ A:(a,b) ϵ ρ for some b ϵ B}

From example2 we see Dom(ρ)={a,c}

☞ Range of a Relation

For any relation ρ from A to B the range is the set of all those elements b ϵ B s.t. (a,b) ϵ ρ for some a ϵ A.

Range(ρ)= {b ϵ B:(a,b) ϵ ρ for some a ϵ A}

From EXAMPLE2 we see Range(ρ)={d,e}

☞ Co-domain of a Relation

For any relation ρ from A to B, B is called the co-domain of ρ.

From EXAMPLE2 we see co-domain of ρ= B= {d,e}

☞ Inverse of a Relation

Let ρ be a relation from A to B, the inverse of the relation ρ denoted by ρ-1 is a relation from B to A and defined by ρ-1 = {(b,a):(a,b)ϵρ}

From EXAMPLE2 we see ρ-1= {(d,a), (e,c)}

☞ Relation on a set

Let S be a non-empty set. A relation ρ on S is a subset of S×S.

☞ Types of Relation on a set 

 ⚫Identity Relation

Let S be a non-empty set. A relation ρ on S is said to be an identity relation if ρ= {(a,a):a ϵ S} 

Example3: S= {1,2,3}

                  ρ= {(1,1), (2,2), (3,3)}    is an identity relation on S.

 ⚫Reflexive Relation

Let S be a non-empty set. A relation ρ on S is said to be reflexive if (a,a) ϵ ρ for all a in S.

Example4: S= {1,2,3}

                 ρ= {(1,1), (2,2), (3,3), (2,3)} is a reflexive relation on S.

 ⚫Symmetric Relation 

A relation ρ on S(non-empty) is said to be symmetric if aρb ⇒ bρa for aϵS ,b ϵ S.

Example5: S= {1,2,3}

                   ρ= {(2,2), (3,1), (1,3)}   is a symmetric relation on S.

 ⚫Transitive Relation

A relation ρ on S(non-empty) is said to be transitive if for any three elements a,b,c in S, (a,b) ϵ ρ and (b,c) ϵ ρ  ⇒(a,c) ϵ ρ.

Example6: S= {1,2,3}

                   ρ= {(1,1), (2,1), (1,2), (2,2)}   is transitive relation on S.

 ⚫Equivalence Relation

A relation ρ on S(non-empty) is said to be an equivalence relation if ρ is reflexive, symmetric, transitive.