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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
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)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 $C \times B$.
Additional Comments
The Cartesian coordinate system arises from the Cartesian product $\mathbb{R}\times\mathbb{R}$.