Products

June 15, 20244 min read

In the last post we encountered the notion of spaces, which are a way of thinking about situations numerically and geometrically. We saw a few examples of spaces:

$\R^n$ is the set of all $n$ -tuples of real numbers, which we think of as $n$ -dimensional space
$\ClsdIntvl ab$ is the interval of numbers between $a$ and $b$ (inclusive on both sides).
in particular, $\I\Defeq\ClsdIntvl01$ consists of all $x$ with $0\le x\le 1$
$S^1$ is the unit circle in $\VecR 2$

In this post, we'll discuss a way to combine two (or more) spaces $X$ and $Y$ into a new space $X\times Y,$ called the product of $X$ and $Y.$ There are two reasons to do this:

it gives us more interesting examples of spaces;
conversely, we understand complicated spaces by expressing them in terms of simpler "building block" spaces. Most questions about $X\times Y$ can be answered by answering the same question for the spaces $X$ and $Y,$ and combining the answers somehow.

The space $X\times Y$ is analogous to the tuple type [X, Y].

Examples

Multiplication

Let's explain why this operation is called multiplication. If $X$ is a finite set, we write $|X|$ for the number of elements of $X,$ also called the cardinality or size of $X.$ For example, $|\{a,b,c\}|=3.$

When $X$ and $Y$ are finite sets, the size of $X\times Y$ is the product of the sizes of $X$ and $Y:$

|X\times Y| = |X||Y|.

For instance, say we have three possible colors and four shapes. The number of possible colored shapes is $3\times4=12:$

\Large \{\red{\mathrm{red}},\, \green{\mathrm{green}},\, \blue{\mathrm{blue}}\} \times \{\blacktriangle,\, \blacksquare,\, \bigstar,\, \bullet\} = \left\{\begin{matrix} \red\blacktriangle&\red\blacksquare&\red\bigstar&\red\bullet\\ \green\blacktriangle&\green\blacksquare&\green\bigstar&\green\bullet\\ \blue\blacktriangle&\blue\blacksquare&\blue\bigstar&\blue\bullet \end{matrix}\right\}

Associativity

One of the most important properties of ordinary multiplication (of numbers) is associativity: this is the property

(ab)c = a(bc).

which means that the expression $abc$ is unambiguous. For example, we could evaluate $2\cdot3\cdot4$ as

\begin{align*} (2\cdot3)\cdot4 &= 6\cdot4 = 24\\ 2\cdot(3\cdot4) &= 2\cdot12 = 24 \end{align*}

and both give the same result.

Something similar is true for the product of spaces, but we have to be more careful. We can give a direct definition of $X\times Y\times Z$ a space of 3-tuples

X\times Y\times Z \Defeq \{(x,y,z) \mid x\in X,\, y\in Y,\, z\in Z\},

or we can build it out of products of two spaces at a time:

\begin{align*} (X\times Y)\times Z &= \{((x, y), z) \mid x\in X,\, y\in Y,\, z\in Z\}\\[.5em] X\times (Y\times Z) &= \{(x, (y, z)) \mid x\in X,\, y\in Y,\, z\in Z\} \end{align*}

These three options aren't "literally" equal. As an analogy, in TypeScript, [X, Y, Z], [[X, Y], Z], and [X, [Y, Z]] are all different types. But there is an obvious way to go between them:

(x,y,z) \longleftrightarrow ((x,y),z) \longleftrightarrow (x,(y,z)).

In math, we generally leave these conversions implicit, so we will write

(X\times Y)\times Z = X\times Y\times Z = X\times(Y\times Z).

I'll say a bit more about this expanded view of equality a few posts from now.

In my area of math (homotopy theory), there's often a lot of work that needs to be done to keep track of all these implicit conversions. One of the difficulties is that when converting between $(X\times X)\times X$ and $X\times(X\times X),$ you don't want to accidentally use the "wrong" conversion

((x,y),z) \longleftrightarrow (z,(y,x))

(which wouldn't typecheck for $(X\times Y)\times Z$ versus $X\times(Y\times Z))$

Products

Examples #

Multiplication #

Associativity #

Examples

Multiplication

Associativity