Ordered Pair

A pair of numbers which always follows a rule that the first components should always be X-component and the second component should always be taken from Y-component, where the pair must be always enclosed within small brackets () and separated by a common is known as an ordered pair.

An ordered pair is always in the form of (x, y). The first component, i.e x-component is also known as the antecedent and the second component, i.e y-component is also called the consequent.

Equality of Ordered Pair

Two ordered pairs are said to be equal if and only if their corresponding components are equal.

i.e. (a,b) = (c,d) if a = c and b = d

Cartesian Product

The Cartesian Product between two sets is the set of all possible ordered pairs with the first element from the first set and second element from the second set. Let A and B be two empty sets. Then the Cartesian product A×B (cartesian product of A and B) read as A cross B is defined as the set of all possible ordered pairs (x, y) such that x∈A and y∈B. Mathematically,

A×B = {(x, y)x∈A and y∈B}

For example,

Let A = {1,2}, B = {2,1}. Prove thatA×B = B×A.

Here, A= {1,2}

B = {2,1}

∴A×B = {(1,2), (1,1), (2,2), (2,1)}

B×A = {(2,1)(2,2)(1,1)(1,2)}

Since all the ordered pairs in A×B and B×A are the same, A×B = B×A.

Ways of representation of Cartesian Product

The cartesian product can be represented in various ways:

1. Set of ordered pairs
   Let A = {1,2} and B= {3,4}.
   The Cartesian product A×B can be represented in the set of ordered pairs as follows:
   A×B = {(1,3), (1,4), (2,3),(2,4)}
   
   
2. Tabulation Method
   Let A = {1,2} and B = {3,4}. The Cartesian product A×B can also be constructed by using the table as follows:
   A×B = {(1,3),(1,4),(2,3),(2,4)}
   
   
3. Tree Diagram
   Let A= {1,2,3} and B = {2,4,6}
   Then, A×B = {(1,2),(1,4),(1,6),(2,2),(2,4),(2,6),(3,2),(3,4),(3,6)}
   The construction of the Cartesian product of this example may be schematically depicted by the following tree diagram.
   
   
4. Mapping/Arrow Diagram
   Let A = {1,2,3} and B = {2,4,6}, then A×B can also be represented as follows:
   So, A×B = {(1,2),(1,4),(1,6),(2,2),(2,4),(2,6),(3,2),(3,4),(3,6)}
   
   
5. Graphical Method
   Let A= {1,2,3} and B = {2,4,6}. Then the Cartesian product A×B can be represented geometrically by the lattice of points.
   In this representation, each point represents an ordered pair of A×B.