Cartesian product of three sets — lesson. Mathematics State Board, Class 10.

PDF chapter test TRY NOW

Definition:

$A$ ,

$B$ and

$C$ are three non-empty sets, then the cartesian product of three sets is the set of all possible ordered triplets given by

$A \times B \times C$

$=$

$\{(a, b, c) \text{ for all } a \in A, b \in B, c \in C\}$ .

Illustration:

Let us consider the following example for the geometrical understanding of the cartesian product of two and three sets.

Consider three sets

$A$ ,

$B$ and

$C$ .

Where

$A$

$=$

$\{2, 3\}$ ,

$B$

$=$

$\{2, 3\}$ and

$C$

$=$

$\{2, 3\}$ .

First, let us find the product

$A \times B$ .

$A \times B$

$=$

$\{2, 3\} \times \{2, 3\}$

$A \times B$

$=$

$\{(2 ,2), (2, 3), (3, 2), (3, 3)\}$

The product

$A \times B$ is geometrically represented as follows:

Here, the product of two sets is represented in the

$xy$

$-$ plane.

The product

$A \times B$ represents the vertices of a square in two dimensions.

Now let us find the product

$A \times B \times C$ .

$A \times B \times C$

$=$

$(A \times B) \times C$ .

$A \times B \times C$

$=$

$\{(2 ,2), (2, 3), (3, 2), (3, 3)\}$

$\times$

$\{2, 3\}$ .

$A \times B \times C$

$=$

$\{(2 ,2, 2), (2 ,2, 3), (2, 3, 2), (2, 3, 3), (3, 2, 2), (3, 2, 3), (3, 3, 2), (3, 3, 3)\}$ .

The product

$A \times B \times C$ is geometrically represented as follows:

Here, the product of three sets is represented in the

$xyz$

$-$ plane.

The product

$A \times B \times C$ represents the vertices of a cube in three dimensions.

Important!

The cartesian product of two non-empty sets provides a shape in

$2$

$-$ dimension, whereas the cartesian product of three non-empty sets provides a shape in

$3$

$-$ dimension.

Previous theory

Exit to the topic

Next theory

Send feedback

Did you find an error?

Send it to us!

5. Cartesian product of three sets

Theory: