Cartesian Product

Formal Definition

GENERAL CASE:
The Cartesian product of sets $S_1, S_2, ..., S_n$ is the set of all ordered $n$-tuples $(a_1, a_2, ..., a_n)$, with $a_i \in S_i$. This is denoted by $S_1 \times S_2 \times ... \times S_n$ or $\prod\limits _{i=1}^n S_i$

CARTESIAN PRODUCT OF TWO SETS
Let $A$ and $B$ be sets. The set

(1)
\begin{align} A \times B=\{(a,b)|a\in A\ and\ b\in B\} \end{align}

is the Cartesian product of $A$ and $B$.

Informal Definition

For two sets, $A$ and $B$, the Cartesian product $A \times B$ is the set containing all unique ordered pairs such that the first entry in each ordered pair is an element of $A$ and the second entry is an element of $B$.

Example(s)

If A={1,2,3} and B={3,4},then we have

(2)
\begin{align} A \times B=\{(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)\}. \end{align}

Non-example(s)

Consider the sets $A = \{1,2,3\}, B = \{a,b\}, C = \{math, cool\}$

The set $\{(a, math), (b, cool)\}$ is NOT the Cartesian product $B \times C$ because it lacks the elements $(a, cool)$ and $(b, math)$.

The set $\{(1,math), (1,cool), (2,math), (2,cool), (3,math), (3,cool), (1, 2, cool)\}$ is NOT the Cartesian product $A \times C$ because it contains the element $(1, 2, cool)$ which is an ordered triple, rather than an ordered pair.

The set $\{(math, a), (cool, b), (math, b), (cool, a)\}$ is NOT the Cartesian product $B \times C$ because the first entry in each ordered pair is an element from $C$ rather than from $B$. This set is actually the Cartesian product $C \times B$.

The Cartesian coordinate system arises from the Cartesian product $\mathbb{R}\times\mathbb{R}$.