Understanding the Cartesian Product of Set A {-1, 1}

This article aims to clarify the process of finding the Cartesian product of a set A {-1, 1} raised to the third power, A × A × A. We will explain the concept step by step and also provide a detailed breakdown of the result.

Defining the Cartesian Product

The Cartesian product of a set A × A × A refers to a set of all possible ordered triples where each element is drawn from the set A. In this case, A consists of two elements: -1 and 1.

Step-by-Step Computation

Let's start by listing the elements of set A:

A {-1, 1}

Now, we proceed to form ordered triples by combining each element of the set A with every other element in A across the three dimensions.

Form Ordered Triples

To form the ordered triples, follow these steps:

Select the first element of the triple. It can be either -1 or the second element of the triple. Again, it can be either -1 or the third element of the triple. It can be either -1 or 1.

This will result in a total of 2 × 2 × 2 8 possible combinations.

Combine the Elements

The possible ordered triples are as follows:

-1 -1 -1-1 -1 1-1 1 -1-1 1 11 -1 -11 -1 11 1 -11 1 1

The result of A × A × A is the set of all possible ordered triples formed by the above combinations. Therefore, the final result is:

A × A × A { -1 -1 -1, -1 -1 1, -1 1 -1, -1 1 1, 1 -1 -1, 1 -1 1, 1 1 -1, 1 1 1 }

Alternative Notations and Representations

Another way to write the Cartesian product A3 might include some alternative notations:

Using ordered triples as follows:

A3 { (1, 1, 1), (-1, -1, -1), (1, -1, 1), (1, 1, -1), (-1, 1, -1), (-1, 1, 1), (-1, -1, 1), (-1, -1, -1) }

Or as a set of elements individually:

A3 {111, -1-1-1, 1-11, 11-1, 1-1-1, -11-1, -1-11, -1-1-1}

Which can also be represented as ordered triples as follows:

A3 { {1, 1, 1}, {-1, -1, -1}, {1, -1, 1}, {1, 1, -1}, {1, -1, -1}, {-1, 1, -1}, {-1, 1, 1}, {-1, -1, 1} }

Conclusion

The computation of A × A × A involves systematically pairing each element from the set A with every other element across three dimensions. The result is a set of 8 unique ordered triples, each containing elements from A.

Understanding the Cartesian product of sets is fundamental in various mathematical and computational applications, including geometry, algebra, and data representation.