The Cartesian product is an operation in mathematics that forms all possible ordered pairs or n-tuples from two or more sets. This operation is a fundamental concept that demonstrates how relationships and combinations between sets can be represented.

Given two sets A and B, their Cartesian product is denoted as A×B and consists of all ordered pairs where a∈A and b∈B. It is the set of all possible ordered pairs formed by taking one element from A and one from B. The term “ordered pair” is crucial, indicating that the sequence matters; thus, typically, (a,b)≠(b,a) where a is chosen from set A and b from set B.

To illustrate this concept with an example:

Let A={1,2,3} and B={d,e}

Then, 

A×B={(1,d),(1,e),(2,d),(2,e),(3,d),(3,e)}

4.1. Properties of the Cartesian Product

1. Ordered Pairs 
The elements in a Cartesian product are ordered pairs, and the order is significant.
For example:
(a,b)≠(b,a) is generally true if A≠B

2. Size of the Cartesian Product
If set A contains mm elements and set BB contains nn elements, then the size of A×BA×B is m×nm×n. For example:

If A={1,2,3} and B={d,e}

Then, 

3×2=6.

3. Generalization: 

The Cartesian product is not limited to two sets; it can be extended to three or more sets. For instance, A×B×C represents the set of all possible ordered triples formed by selecting one element from each set.