The Cartesian Product

The Cartesian product (or Cross product) of two sets $A$ and $B$ is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

If $A$ and $B$ are two sets, their Cartesian product, denoted $A \times B$, is defined as follows:

\[ A \times B = \{ (a, b) \mid a \in A \text{ and } b \in B \} \]

Here, $a$ is an element of set $A$ and $b$ is an element of set $B$.

Example

Consider two simple sets:

$A = \{1, 2\}$

$B = \{x, y\}$

The Cartesian product $A \times B$ will be:

\[ A \times B = \{(1, x), (1, y), (2, x), (2, y)\} \]

Imagine set $A$ represents the rows of a table, and set $B$ represents the columns.

The Cartesian product $A \times B$ can be visualized as a table where each cell contains a pair $(a, b)$.

\[
\begin{array}{c|c|c}
    & x & y \\
    \hline
    1 & (1, x) & (1, y) \\
    2 & (2, x) & (2, y) \\
\end{array}
\]

In geometry, the Cartesian plane is an example of the Cartesian product $\mathbb{R} \times \mathbb{R}$, where each point is a pair of coordinates $(x, y)$.

Extension to Multiple Sets

The concept of the Cartesian product can be extended to more than two sets.

For example, the Cartesian product of three sets $A$, $B$, and $C$ is the set of all ordered triples $(a, b, c)$ where $a \in A$, $b \in B$, and $c \in C$:

\[ A \times B \times C = \{ (a, b, c) \mid a \in A, b \in B, c \in C \} \]

When considering the Cartesian product of more than two sets, the concept naturally extends.

Formally, if there are $n$ sets, the Cartesian product is the set of all ordered $n$-tuples $(a₁, a₂, ... a_n)$.

$$ A_1 × A_2 × ... × A_n \ = \ \{ \ (a_1, a_2, ..., a_n) \ \ \ | \ \ \ a_1 ∈ A_1, a_2 ∈ A_2, ..., a_n ∈ A_n \} $$

In general, if there are $n$ sets, each element of the Cartesian product is called an n-tuple.

Let's look at a numerical example with three sets:

$$ A = \{ 1, 2 \} $$

$$ B = \{ 3, 4 \} $$

$$ C = \{ 5, 6 \} $$

The Cartesian product $A \times B \times C$ is the set of all ordered triples formed by taking one element from $A$, one from $B$, and one from $C$.

$$ A × B × C = \{ (1, 3, 5), (1, 3, 6), (1, 4, 5), (1, 4, 6), \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2, 3, 5), (2, 3, 6), (2, 4, 5), (2, 4, 6) \} $$

In this case, we have combined each element of the first set with each element of the second and third sets, creating all possible combinations of three elements.