Ordered Pairs and Cartesian Product


   
 
Representation of a Relation
(i) Roster form
 
(ii) Set builder form
 
(iii) By tables
 
(iv) Arrow diagram
 
(v) By graphs
 
(i) Roster form:
 
The ordered pairs are listed,
 
R = {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5)}
 
Where A = {1, 2, 3, 4, 5} and R means 'is twice' in A x A
 
xRy means x = 2y.
 
 
(i) x = 2y is called the defining sentence because it defines the relation between x and y.
 
(ii) Set builder form: We state the rule which defines the relation of the ordered pairs (x, y). It is expressed as {(x, y) : x A, y B defining sentence}. The defining sentence states the rule of the relation xRy.
 
{(x, y) : x, y N x + y = 5}
 
x + y = 5 is called the defining sentence.
 
This relation can be expressed in roster form as {(1, 4), (2, 3), (3, 2), (4, 1)}.
 
(iii) By tables: The relation can be expressed by a table form. Consider an example.
 
(a)
 
 
(iv) Arrow diagram and
 
(v) By graphs, we have discussed in the previous topic 'the ordered pairs'.
 
 
Find the domain and the range, given
 
R = {(x, y) : y = 3x, x N, 6 < x 10}
 
 
R = {(7, 21), (8, 24), (9, 27), (10, 30)}
 
Domain = {7, 8, 9, 10} and Range = {21, 24, 27, 30}
 
 
Given A = {3, 4, 5}, R = {(x, y) A x A : x = y - 1}
 
(i) List the ordered pairs of R.
 
(ii) List the elements of the domain of R.
 
(iii) List the elements of the range of R.
 
 
A x A = {(3, 3), (3, 4), (3,5), (4, 3), (4, 4), (4, 5), (5, 3), (5, 4), (5, 5)}
 
(i) R = {(3, 4), (4, 5)}
 
(ii) Domain = {3, 4}
 
(iii) Range = {4, 5}
 
  
     
   
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