### Theory:

Let us learn the cartesian product of two sets in this article.
Illustration:
Let us consider the sets $$A$$ and $$B$$.

Where, $$A$$ is the set boys named $$1$$, $$2$$ and $$3$$.

That is, $$A$$ $$=$$ $$\{1, 2, 3\}$$.

And $$B$$ is the set of girls named $$a$$, $$b$$ and $$c$$.

That is, $$B$$ $$=$$ $$\{a, b, c\}$$.

Let us see the possible ways of pairing each boy with a girl for the dance program using an arrow diagram.

We can select $$9$$ distinct pairs as given below:

 $$(1,a), (1,b), (1,c), (2,a), (2,b), (2,c), (3,a), (3,b), (3,c)$$

Here, these pairs represent the cartesian product of the set of boys and set of girls.

Based on the above illustration, we will define the Cartesian product of the sets $$A$$ and $$B$$ as follows:
Definition:
If $$A$$ and $$B$$ are two non-empty sets, then the set of all ordered pairs $$(a, b)$$ such that $$a \in A$$, $$b \in B$$ is called the Cartesian Product of $$A$$ and $$B$$, and is denoted by $$A \times B$$.

Thus, $$A \times B$$ $$=$$ $$\{(a, b) | a \in A, b \in B\}$$.
Important!
• $$A \times B$$ is the set of all ordered pairs between the elements of the sets $$A$$ and $$B$$ where the first ordinate belongs to set $$A$$ and the second ordinate belongs to set $$B$$.
• $$B \times A$$ is the set of all ordered pairs between the elements of the sets $$A$$ and $$B$$ where the first ordinate belongs to set $$B$$ and the second ordinate belongs to set $$A$$.
• If $$a = b$$, then $$(a, b) = (b, a)$$.
• For any three sets $$A$$, $$B$$ and $$C$$, the following properties are true.
1. $$A \times (B \cup C)$$ $$=$$ $$(A \times B) \cup (A \times C)$$ (Distributive property over union).
2. $$A \times (B \cap C)$$ $$=$$ $$(A \times B) \cap (A \times C)$$ (Distributive property over intersection).