Cartesian Product of three Set - Illustration for Geometrical understanding
Cartesian Product of three Sets
If A, B, C are three non-empty sets then the cartesian product of three sets is the set of all possible ordered triplets given by
$$ A \times B \times C = \{ (a,b,c) \mid a \in A, b \in B, c \in C \} $$
Illustration for Geometrical understanding of cartesian product of two and three sets
Let's define our sets:
Set A
{0, 1}
Set B
{0, 1}
Set C
{0, 1}
Step 1: Find the Cartesian Product of A and B (A × B)
{0, 1} × {0, 1}
Result (A × B):
{(0, 0), (0, 1), (1, 0), (1, 1)}
Representing A×B in the xy - plane we get a picture shown in Fig. 1.5.
(A×B)×C = {(0, 0),(0,1),(1, 0),(1,1)} × {0,1}
= {(0, 0, 0),(0, 0,1),(0,1, 0),(0,1,1),(1, 0, 0),(1, 0,1)(1,1, 0),(1,1,1)}
Representing A×B ×C in the xyz - plane we get a picture as shown in Fig. 1.6
Thus, A×B represent vertices of a square in two dimensions and A×B ×C represent vertices of a cube in three dimensions.
NOTES
In general, cartesian product of two non-empty sets provides a shape in two dimensions and cartesian product of three non-empty sets provide an object in three dimensions.