Set product

by Patrick Stevens Aug 25 2016 updated Aug 26 2016

A fundamental way of combining sets is to take their product, making a set that contains all tuples of elements from the originals.

[summary: The product of two sets $~$A$~$ and $~$B$~$ is just the collection of ordered pairs $~$(a,b)$~$ where $~$a$~$ is in $~$A$~$ and $~$b$~$ is in $~$B$~$. The reason we call it the "product" can be seen if you consider the set-product of $~$\{1,2,\dots,n \}$~$ and $~$\{1,2,\dots, m \}$~$: it consists of ordered pairs $~$(a, b)$~$ where $~$1 \leq a \leq n$~$ and $~$1 \leq b \leq m$~$, but if we interpret these as integer coordinates in the plane, we obtain just an $~$n \times m$~$ rectangle.]

[summary(Technical): The product of sets $~$Y_x$~$ indexed by the set $~$X$~$ is denoted $~$\prod_{x \in X} Y_x$~$, and it consists of all $~$X$~$-length ordered tuples of elements. For example, if $~$X = \{1,2\}$~$, and $~$Y_1 = \{a,b\}, Y_2 = \{b,c\}$~$, then $$~$\prod_{x \in X} Y_x = Y_1 \times Y_2 = \{(a,b), (a,c), (b,b), (b,c)\}$~$$ If $~$X = \mathbb{Z}$~$ and $~$Y_n = \{ n \}$~$, then $$\prod{x \in X} Yx = {(\dots, -2, -1, 1, 0, 1, 2, \dots)}]

[todo: define the product as tuples]

[todo: several examples, including R^n being the product over $~$\{1,2, \dots, n\}$~$; this introduces associativity of the product which is covered later]

[todo: product is associative up to isomorphism, though not literally]

[todo: cardinality of the product, noting that in the finite case it collapses to just the usual definition of the product of natural numbers]

[todo: as an aside, define the product formally in ZF]

[todo: link to universal property, mentioning it is a product in the category of sets]