An ordered pair is a pair of elements, where the order of elements is definite. It is written by enclosing the pair of elements in brackets as ( x, y ). In an ordered pair the two members need not be different. For example ( 7, 7 ).

If ( x₁ ; y₁ ) and ( x₂, y₂ ) represent ordered pairs then ( x₁, y₁) = ( x₂, y₂ ) if and only if x₁ = x₂ and y₁ = y₂. But ( 5, 9 ) ¹
( 9, 5 ).

Let A and B be two sets. The cartesian product of two sets A and B is the set of all ordered pairs in which the first element of every pair belongs to the set A and the second belongs to set B. It is written as A ´ B and read as A cross B.

Example If A = { 1, 2, 3 }, B = { 4, 5, 6 } then find A ´ B and B ´ A

Solution : A ´ B = { ( 1, 4 ), ( 1, 5 ), ( 1, 6 ), ( 2 , 4 ),
( 2, 5 ), ( 2 , 6 ), ( 3, 4 ), ( 3, 5 ), ( 3, 6) }

Clearly A ´ B ¹ B ´ A

Example If A = { x | x Î N Ù x £ 3 } , write A ´ A and graph it.

Solution : A = { x | x Î N Ù x £ 3 } = { 1, 2, 3 }

Therefore A ´ A = { 1, 2, 3 } ´ { 1, 2, 3 }