Direct Products

Why

We generalize the product of two sets to a product of a family of sets. To do so we discuss sets of families.

Discussion for pairs

Suppose $X$ and $Y$ are nonempty sets. There is a natural correspondence between the product $X \times Y$ (see Set Products) and the set of families

\[ Z = \Set*{z: \set{i, j} \to (A \cup B)}{z_i \in A \text{ and } z_j \in B} \]

where $\set{i,j}$ is any unordered pair with $i \neq j$.

The set $Z$ can be put in one-to-one correspondence with $X \times Y$. The family $z \in Z$ corresponds with the pair $(z_i, z_j)$. The pair $(a, b)$ corresponds to the family $z \in Z$ defined by $z(i) = a$ and $z(j) = b$. So, ordered pairs can be put in one-to-one correspondence with families. The generalization of Cartesian products to more than two sets generalizes the notion for families.

Definition

Suppose $\set{A_i}_{i \in I}$ is a family of sets. The direct product ( or Cartesian product, family Cartesian product) of $A$ is the set of all functions (i.e., families) $a: I \to X$ which satisfy $a_i \in A_i$ for every $i \in I$.

A function on a product is called a function of several variables and, in particular, a function on the product $X \times Y$ is called a function of two variables.

Notation

We denote the product of the family $\set{A_i}_{i \in I}$ by

\[ \textstyle \prod_{i \in I} A_i \]

We read this notation as “product over i in I of A sub-i.” Other notation in use includes $\times _{i \in I} A_i$.

Projections

The word “projection” is used in two senses with families. Let $I$ be a set, and let $\set{A_i}_{i \in I}$ be a family of sets. Define $A = \prod_{i \in I} A_i$.

First, let $J \subset I$. There is a natural correspondence between the elements of $A$ and those of $\prod_{j \in J} A_j$. To each element $a \in A$, we restrict $a$ to $J$ and this is restriction is an element of $\prod_{j \in J} A_j$. The correspondence is called the projection of $A$ onto $\prod_{i \in J} A_i$. The projection in this sense is a set of families.

Second, consider the value of a family $a \in A$ at $j$. We call $a_j$ the projection of $a$ onto index $j$ or the $j$-coordinate of $a$. This word coordinate is meant to follow the language used in defining ordered pairs. The projection in this sense is an element of $A_j$. The $j$th projection is a function mapping $\prod_{i \in I} X_i$ to $X_j$.