CLASS-11
RELATION & FUNCTION - CARTESIAN PRODUCT

Cartesian Product –

Consider a set B of two boys and another set G of four girls.

     B = {Smith, Richard},          G = {Viva, Mona, Monalisa, Richa}

The possible number of ways of forming partnership for a mixed doubles badminton team from these two sets is equal to the possible number of ordered pairs that can be formed from the set of two boys and the set of 4 girls. The set of all these possible ordered pairs is called the Cartesian Product of the given sets.

In the above mentioned illustration, the possible partnerships are – (Smith, Viva), (Smith, Mona), (Smith, Monalisa), (Smith, Richa), (Richard, Viva), (Richard, Mona), (Richard, Monalisa), (Richard, Richa).

The Cartesian product of sets B and G, denoted by B X G = {(Smith, Viva), (Smith, Mona), (Smith, Monalisa), (Smith, Richa), (Richard, Viva), (Richard, Mona), (Richard, Monalisa), (Richard, Richa)}

Let A & B be two non-empty sets. Then the set of all possible ordered pairs (x, y) such that the first competent x of the ordered pairs is an element of A, and the second component y is an element of B, is called the cartesian product of sets A & B. It is denoted by A X B which reads “A cross B”.

Thus, A X B = {(a, b) ǀ a ∈ A and b ∈ B}

Also, n(A X B) = n(A). m(B).

If set A has n elements and set B has m elements then the product set A X B has nm elements.

Please Note.1) – The Cartesian product A X B is not the same as B X A. In A X B, the set A is named first and so its elements will appear as the first components of the ordered pairs. In B X A, the set B is named first so in this case its elements will appear as the first components of the ordered pairs.

Please Note.2) – If either A or B is the null set, then we define A X B to be the null set. For example, if A = {a, b}, and B = ϕ, then A X B = ϕ.

Please Note.3) – If either A or B is an infinite set and other is a non-empty set, then A X B is also an infinite set.

Please Note.4) – A X ϕ is the empty set where A is any set.

Please Note.5) – If A and B are two non-empty sets having n elements in common, then A X B, B X A have n² elements in common.

Please Note.6) - If A = B, then A X B becomes A X A and is denoted by A²

Example.1) If A = {a, b}, and B = {1}, then find A X B

Ans.) A X B = {(a, 1), (b, 1)} and B X A = {(1, a), (1, b)}

Example.2) If A = {2, 4, 6}, then find A²

A²= A X A = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)}

Example.3) If  A X B = {(a, x), (a, y), (b, x), (b, y)}, find A & B.

Ans.) Given, A X B = {(a, x), (a, y), (b, x), (b, y)}

                                    = (a, b) (x, y)

=> A = {a, b},  B = {x, y}