LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

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}**