Relations

       A relation between two sets A and B is any subset of A× B . 

Symbolically, R is a relation from A to B if R⊂ A ×B. 

Note

    (1)If A = B , that is , a relation from A to A is called a relation on A . 

    (2)If R is a relation from A to B and (a, b) ∈R , then we also express this fact by writing aRb and say a is R-related to b . 

Number of different relation from A to B

     If A consists of m elements and B consists of n elements, then A×B consists  of mn elements and so 

  P(A×B) will have 2^mn elements. Hence, the total number of different relations from A to B is 2^mn. 

Domain and Range of a Relation

    If R is a relation from a set A to another set B , then the domain of R is the set of all first coordinates of the members of R and the range of R is the set of all second coordinates of the members of R. Thus,  

         Domain of R = { a:(a ,b ) ∈R for some b∈ B }. 

         Range of R = { b: (a, b ) ∈  R for some a∈ A }. 

          Domain                       Range

The Inverse Relation 

    Let R be a relation from a set A to another  set B . Then the inverse relation R^-1 of R is defined by setting (b ,a) ∈ R^-1 iff (a, b) ∈R. 

Evidentially , the range of R is the domain of R^-1 and the domain of R is the range of

R^-1 . 

Note --  ( R^-1)^-1 = R 

Example –Let A = { a, b ,c } ,B= { 1,2,3 }  

              And    R = { (a, 1), (a,3) , (b,3),(c,3) } , then inverse of R is  

                         R^-1 = { (1,a),(3,a),(3,b),(3,c) } 

 Various Types of Relations

    (1)Identity Relation 

Let A be any set . Then the identity relation on A is defined by setting

(a , b) ∈ I ,if a= b . 

The domain and range of I are both A. 

Or the relation  

              I = { (a, a): a∈ A } is called the identity relation on A. 

Example –

       If A = { 1,2,3,4 } , then identity relation  

       I = { (1,1) ,(2,2) ,(3,3),(4,4) } 

   (2)Reflexive Relation

A relation R is said to be reflexive relation , if every element of set A is related to itself .Thus , if (a ,a) ∈ R , for all a ∈ A  , then R is reflexive . 

    (3)Symmetric Relation

A relation R is said to be symmetric relation if whenever   

                 ( a, b ) ∈ R , then    (b, a) ∈ R 

    (4)Transitive Relation

A relation R is said to be an transitive relation if whenever (a ,b ) ∈ R , (b , c) ∈ R, then (a, c) ∈R. 

    (5)Equivalence Relation

A relation R is called an equivalence relation if it is  

(a)Reflexive  

(b)Symmetric  

(c)Transitive  

Note

1. R is reflexive  ⇒ R^-1 is reflexive. 

1. R is symmetric ⇒R^-1 is symmetric. 

1. R is transitive ⇒R^-1 is transitive. 

Examples

    (1)If ( x +y , 1) and ( 3 , x -y ) are equal , then find the value of x and y . 

Sol.         We have , ( x + y , 1) = ( 3 , x -y )  

Thus ,          x + y = 3 and x -y = 1 

Solving these two equations , we have x = 2 , y = 1 

    (2) If A = {1 } and B = { 2,3} , then find A× B . 

Sol.      A×B = { ( 1,2) , (1,3) } 

    (3)If R = { ( 1,3 ) , ( 1,5 ) , (3,2) , (2,5 ) } , then find domain and range of a relation R . 

Sol.          Domain of R = { 1,2,3 } 

                 Range of R = { 2,3,5 } 

    (4)Let R be the relation on the set of real numbers defined by “ x is less than y where x ,y ∈R.” 

         Then , R^-1 is the relation defined by “ x is greater than y where x ,y ∈R “. 

    (5) If A = { 1,2,3 } , then  

           R1 = { (1,1),(2,2),(3,3),(1,3) } is reflexive relation . 

           R2= { ( 1,1),(2,2) } is not reflexive because (3,3) does not belongs to R2. 

           R3 = { (1,2),(2,1) } is symmetric but not reflexive . 

           R4 = { (1,1),(1,2) } is neither reflexive nor symmetric , but is transitive . 

#Let A be the set of all lines in a plane. 

   Let R⊆ A×A where  

     R = { l , m) : l , m ∈ A , l || m } , then R is an equivalence relation. 

#The relation of equality on integers is an equivalence relation. 

# Let Z = set of integers

 then the usual ≤ is not an equivalence relation on Z as it is  

Reflexive: as a ≤ a  for all a∈ Z . 

Transitive: as a ≤ b , b ≤ c  ⇒ a ≤ c. 

But not symmetric: as a ≤ b , b ≤ a ⇒ a= b.