# CLASS-11GENERALIZATION 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.