nLab
cartesian product

The cartesian product is a product in Set, the category of sets.

Definition

Given any family $(A_i{)}_{i:I}$ of sets, the cartesian product ${\prod }_i{A}_i$ of the family is the set of all functions $f$ from the index set $I$ with $f_j$ in $A_j$ for each $j$ in $I$.

As stated, the target of such a function depends on the argument, which is natural in dependent type theory; but if you don’t like this, then define ${\prod }_i{A}_i$ to be the set of those functions $f$ from $I$ to the disjoint union ${⨄}_i{A}_i$ such that $f_j\in A_j$ (treating $A_j$ as a subset of ${⨄}_i{A}_i$ as usual) for each $j$ in $I$.

In traditional forms of set theory, one can also take the target of $f$ to be the union ${\bigcup }_i{A}_i$ or even the class of all objects (equivalently, leave it unspecified).

Special cases

Given sets $A$ and $B$, the cartesian product of the binary family $(A,B)$ is written $A\times B$; its elements $(a,b)$ are called ordered pairs. (In set theory, one often makes a special definition for this case, defining

$$(a,b)=\{\{a\},\{a,b\}\}$$(a,b) = \{\{a\},\{a,b\}\}

rather than as a function so that ordered pairs can then be used in the definition of function. From a structural perspective, however, this is unnecessary.)

Given sets $A_1$ through $A_n$, the cartesian product of the $n$-ary family $(A_1,\dots ,A_n)$ is written ${\prod }_{i=1}^n{A}_i$; its elements $(a_1,\dots ,a_n)$ are called ordered $n$-tuples.

Given sets $A_1$, $A_2$, etc, the cartesian product of the countably infinitary family $(A_1,A_2,\dots )$ is written ${\prod }_{i=1}^{\mathrm{\infty }}{A}_i$; its elements $(a_1,a_2,\dots ,)$ are called infinite sequences.

Given a set $A$, the cartesian product of the unary family $(A)$ may be identified with $A$ itself; that is, we identify the ordered singleton $(a)$ with $a$.

The cartesian product of the empty family $()$ is the point, a set whose only element is the empty tuple $()$; we often call this set $1$ (or $pt$, when we're Urs) and write its element as $*$.