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GENERALIZATION OF CARTESIAN PRODUCT OF MORE THAN TWO SETS

**GENERALIZATION OF CARTESIAN PRODUCT OF MORE THAN TWO SETS -**

**Ordered Triplet –**

**If there are 3 sets A, B, C, then
choosing 3 elements a, b, c such that a ∈ A, b ∈ B,
c ∈ C. we form an ordered triplet (a, b, c). The set of all such ordered
triplets is called the Cartesian product of the three sets A, B, C, and is
denoted by A X B X C.**

__Definition__ – For any non-empty
set A, we define

** (A X A X A) = {(a, b,
c) : a, b, c ∈ A}**

**Three numbers a, b, c are listed
in a specific order and enclosed in parentheses. Then, **

**(3, 4, 5) ≠ (4, 3, 5) ≠ (5, 4, 3), etc.,**

**Similarity, if there are ‘n’
sets A₁, A₂, A₃,…………..Aₑ**

**Example.1) If A = {-1, 1}, find A X
A X A**

**Ans.) A X A = {-1, 1} X {-1, 1}**

** = {(-1, -1), (-1, 1),
(1, -1), (1, 1)}**

**A X A X A = {(-1, -1), (-1, 1),
(1, -1), (1, 1)} X {-1, 1}**

** = {(-1, -1, -1), (-1, 1, -1), (1, -1, -1), (1,
1, -1), (-1, -1, 1), (-1, 1, 1), ( 1, -1, 1), (1, 1, 1)} **

**Cartesian product of the set of
reals with itself.**

**Example.2) Let R be the set of real
numbers. What does (R X R X R) represent ?**

**Ans.) R X R X R = {(a, b, c) :
a, b, c ∈ R}**

**Thus, (R X R X R) represents the
set of all coordinates of points in three dimensional plane.**

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