Cartesian Products

Ordered $n$-tuple

The ordered $n$-tuple $(a_1,a_2,\dots ,a_n)$ is the ordered collection that has $a_1$ as its first element, $a_2$ as its second element, ... , and $a_n$ as its $n$-th element

In particular, ordered $2$-tuples are called ordered pairs.

Cartesian Product

Let $A$ and $B$ be sets. The Cartesian product of $A$ and $B$, denoted by $A\times B$, is the set of all ordered pairs $(a,b)$, where $a\in A$ and $b\in B$. Hence

$$A\times B=\left\{(a,b)\mid a\in A\wedge b\in B\right\}$$

The Cartesian product of the sets $A_1,A_2,\dots ,A_n$, denoted by $A_1\times A_2\times \cdots \times A_n$ is the set of ordered $n$-tuples $(a_1,a_2,\dots ,a_n)$, where $a_i$ belongs to $A_i$ for $i=1,2,\dots ,n$. In other words

$$A_1\times A_2\times \cdots \times A_n=\left\{(a_1,a_2,\dots ,a_n)\mid a_i\in A_i\text{ for }i=1,2,\dots ,n\right\}$$

Relation

A subset $R$ of the Cartesian Product $A\times B$ is called a relation from set $A$ to the set $B$. A relation from $A$ to itself is called a relation on $A$

See Equivalence Relation