Tuples

Ordered Pairs

In 1921, the Polish logician Kazimierz Kuratowski offered the following definition:

This is one of the greatest examples of "working with what you've got" and the extent of mathematical cleverness. Kuratowski essentially said: "Why are we so concerned about which goes first and which goes second? It's unnecessary." And he had a point. The notion of "first" and "second" is entirely relative.

Suppose we had two objects side by side.

If we read from left to right, ${a}$ is first and ${b}$ is second. If we read from right to left — a perfectly permissible thing to do — then ${b}$ is first and ${a}$ is second. If we read it from top to bottom — again perfectly permissible — then, well, all bets are off. Kuratowski was astute enough to realize that searching for a way to define "first" and "second" was a zero sum game. Even if we managed to establish a rigorous definition for "first" and "second," the situation wouldn't change because there would always be an alternative view.

To Kuratowski, the notions of "first" and "second" are just convenient ways of achieving what we're really after: A way of communicating the idea that

is different from

Thus, we don't need a way to establish what's first or second. What we need is a way to distinguish between two sets. If we look at Kuratowski's definition, it does exactly that. Let's say ${a = 0}$ and ${b = 1.}$ Then, by Kuratowski's definition, we have:

Now let's say ${a = 1}$ and ${b = 0.}$ Then we have:

Looking at the two sets, we see that they are, in fact, different:

Because we have a way of distinguishing between ${(a,b)}$ and ${(b,a),}$ the notion of "first" and "second" is just a cherry on top, and it just so happens that we prefer interpreting ${(a,b)}$ as "${a}$ is first, and ${b}$ is second."

An added benefit of Kuratowski's definition is that it allows us to state when two ordered pairs are equal:

Tuples

Ordered pairs allow us to express a pair ${(a,b).}$ But what if we want to express something like:

Well, we can take a similar approach to Kuratowski's. But instead of nesting a set, we'll nest an ordered pair:

Again, what's important is that we have a way of distinguishing the set represented by the expression above and some other set. If we expanded the above expression, we'd get:

Once more, we still have a way to differentiate between sets. Suppose ${a=1,}$ ${b=2,}$ and ${c=3.}$ Then:

Now suppose ${a=1,b=3,c=2.}$ Then:

Those are very clearly different sets. The fact that we're simply nesting ordered pairs allows us to extend the same idea to ${n}$ elements.

We call the construct above an ${n}$-tuple. Notice that this effectively generalizes the notion of an ordered pair — an ordered pair is simply a 2-tuple.

The Cartesian Product

With the notion of tuples, we can now represent a wide variety of objects mathematically. Suppose we have two sets:

Now that we have tuples, we can create a set that contains all the possible 2-tuples ${(a,b),}$ where ${a \in A}$ and ${b \in B.}$ Let's explicitly lay out that set:

More explicitly:

Because we have the notion of ${n}$-tuples, we can also define the Cartesian product generally: