# Set product

https://arbital.com/p/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]