### Theory:

Illustration 1:
Consider the sets $$A$$ and $$B$$.

Let $$A$$ $$=$$ $$\{2, 3\}$$ and $$B$$ $$=$$ $$\{1, 2, 3\}$$.

We shall write the product $$A \times B$$ as follows:

$$A \times B$$ $$=$$ $$\{(2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)\}$$

The graphical illustration of $$A \times B$$ is given by:

Similarly, we will write the product $$B \times A$$ as follows:

$$B \times A$$ $$=$$ $$\{(1, 2), (1, 3), (2, 2), (2, 3), (3, 2), (3, 3)\}$$

The graphical illustration of $$B \times A$$ is given by:

From the graphs of $$A \times B$$ and $$B \times A$$, it is observed that $$A \times B$$ $$\neq$$ $$B \times A$$.

Important!
• In general, $$A \times B$$ $$\neq$$ $$B \times A$$ but $$n(A \times B)$$ $$=$$ $$n(B \times A)$$.
• $$A \times B = \phi$$ if and only if $$A = \phi$$ or $$B = \phi$$.
• If $$n(A) = x$$ and $$n(B) = y$$, then $$n(A \times B) = xy$$.
Illustration 2:
Consider the sets $$A$$ and $$B$$.

Let $$A$$ $$=$$ $$\{\text{Set of all numbers in the interval } [2, 5]\}$$ and $$B$$ $$=$$ $$\{\text{Set of all numbers in the interval } [4, 5]\}$$.

The product $$A \times B$$ corresponds to the intersecting region of the given intervals. In other words, it the set of all points $$(x,y)$$ lying within the common region.

We shall represent the product $$A \times B$$ graphically as follows:

The product $$A \times B$$ corresponds to the middle rectangular region. That is, it consists of all points $$(x, y)$$ within this rectangular region.

Important!
The product $$\mathbb{R} \times \mathbb{R}$$ is called the cartesian plane where it represents the set of all points $$(x, y)$$ where $$x$$, $$y$$ are real numbers.