Fundamental Concepts of Ordered Pairs and Relations

There are some operations which when performed on two sets give rise to another set. We will now define certain operations on sets and examine their properties. Henceforth, we will refer all our sets as subsets of some universal set.

Given two non-empty sets P and Q. The cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e., P × Q = { (p,q) : p  ∈ P, q  ∈ Q } If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = Ø

Suppose A is a set of 2 colours and B is a set of  3 objects, i.e., A = {red, blue}and B = {b, c, s}, where b, c and s represent a particular bag, coat and shirt, respectively.
From the illustration given above we note that A × B = {(red,b), (red,c), (red,s), (blue,b), (blue,c), (blue,s)}. 
Note:
(i) Two ordered pairs are equal, if and only if  the corresponding first elements are equal and the second  elements are also equal.
(ii) If there are p elements in A and q elements in B, then there will be pq elements in A × B, i.e.,  if n(A) = p and n(B) = q,  then n(A × B) = pq. 
(iii) If A and B are non-empty sets and either A or B is an infinite set, then so is A × B. 
(iv) A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product  A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The second element is called the image of  the first element.

The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.

The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R. Note that range ⊂ codomain.

(i) A relation may be represented algebraically either by the Roster method or by the Set-builder method. 
(ii) An arrow diagram is a visual representation of a relation.
Consider the two sets P = {a, b, c} and Q = {Ali, Bhanu, Binoy, Chandra, Divya}. 

The cartesian product of P and Q has 15 ordered pairs which can be listed as P × Q = {(a, Ali), (a,Bhanu), (a, Binoy), ..., (c, Divya)}. We can now obtain a subset of P × Q by introducing a relation R between the first element x and the second element y of each ordered pair (x, y) as R= { (x,y): x is the first letter of the name y, x ∈ P, y ∈ Q}. Then R = {(a, Ali), (b, Bhanu), (b, Binoy), (c, Chandra)} A visual representation of this relation R (called an arrow diagram) is shown in Fig .
 A relation R from A to A is also stated as a relation on A.