Relations

Describing a relation:
1. List form or Roster form: In this method we list all the ordered pairs that satisfy the formula given in the relation.
Example :- i) E is the relation having the property “is equal to” for its elements.
Solution: Given that E is the relation consisting all those ordered pairs whose first coordinates are equal to the second coordinates there fore
    E = {(1, 1),(2, 2),(3,3)}
2. Set builder form of a reaction:- In these method we subscribe the relation by stating the property that connects the first and second coordinates of every pair of the relation.
Example: Write the relation E, L and G in A = {1, 2, 3} described in previous example in the set builder form.
Sol: i) E = {(1, 1),(2, 2),(3, 3)} = \( \{ (x,y)/(x,y) \in A \times A,x = y\} \)

Inverse relation:
Example 1: If R = {(2, 3),(2, 4),(3, 4),(4, 3),(3, 2),(4,2)}is a relation A = {2, 3, 4}.
    Find R^–1 
    R^–1 ={(3, 2), (4, 2), (4, 3), (3, 4),(2, 3), (2, 4)}
    Observe that R = R^-1

Types of relations:
1. One – One relation : A relation R : A\( \to \)B is said to be one – one relation if no two elements of A have the same Image in B
Eg:

2. One to many relation: A relation R : A\( \to \)B is said to be one – to – many relation if an element of A is related to two or more elements of b
Eg:

3. Many – one Relation: A relation R : A\( \to \)B is said to be many – one relation if two or more elements of A are related to an element of B
Eg:

4. Many – Many relation: A relation R : A\( \to \)B is said to be many – many relation if two or more elements of A are related to two or more elements of B
Eg: