In this little post, we'll expand on what we've learned about sets and functions. Specifically, we'll double-down on the claim that sets are vital to everything we do in mathematics. Functions are not just a way to desribe interactions between sets: functions are sets!

Many people will be familiar with the idea of an ordered pair. This is often how we describe "points" created by a function. When we write the ordered pair in the context of an existing function , we typically mean In this way, a function can be completely diescribed by its set of ordered pairs:

This "set-builder" notation is new. Let's describe it briefly.

Now, we have a way to describe a function as a set. Namely, a function is just a set where every element is an ordered pair; each ordered pair describes a rule of the form

This is a great first step! We are back to sets being the backbone of everything. Yet, there is something missing. All we've talked about are sets, and all of a sudden we introduce this "ordered pair." What are the rules about an ordered pair? What is it? As far as we can tell, it's some arbitrary object we created, unrelated to sets. As you may recall, a set is unordered. In set notation we write So, how do ordered pairs relate to sets? Do they?

This will be our first dive into greater abstraction, analyzing an object we are intuitively familiar with in the context of another object we know as a foundation. In particular, we want to move from intuition to definition. To start, what is the defining aspect of an ordered pair?

As the name suggests, it is a pair of objects that happens to be in a certain order. Specifically, if we have two ordered pairs and , what does it mean to have Intuitively, we would say "each of the parts of the pair are equal to the part in the othe pair." As a definition though, we would say means and

This is our defining rule for an ordered pair. Next, we need to figure out how to best describe this relationship using sets.

Here is the accepted mathematical definition.

This strikes most people as quite strange at first. But as we consider the implications of this definition, it eventually gives us an idea of "order" we want from an ordered pair. The first element is , the one that exists in both sets. The second element is the element that remains.

The payoff is we now have a full set-based definition of a function. We saw how a function is a set of ordered pairs, and we now see that an ordered pair is just a set as well. This set just happens to contain two more sets!

Here is our fully set-based definition of a function.

When doing mathematics, most people do not concern themselves with this definition in their daily work. It is good to see, to understand the foundation of what we do, but we come up with alternate notation like to make things a bit easier to communicate.

We can go even deeper, writing everything in terms of various logical statements. All of math can theoretically be written down using a symbol soup. But that is ultimately unhelpful for communication, and having a clearly communicated idea is one of the beauties of well-formulated mathematics.