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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 $\mathbb{R}\times\mathbb{R}$.