FMCC Going Deeper: Sets and Solution Sets

Section 2.5 Going Deeper: Sets and Solution Sets

In mathematics, a set is just a collection of objects. In a very real way, sets are among the most fundamental objects in mathematics. There is a particular type of set that is associated with equations involving variables, which are known as solution sets. Basically, a solution set is the set of all the numbers that you can plug into the equation to get a valid if-then statement regarding the values.

For example, the solution set of the equation $x^2 = 4$ is $\{ -2, 2 \}\text{.}$ We can check this b thinking through the appropriate if-then statements:

If $x = -2\text{,}$ then $x^2 = 4\text{.}$
If $x = 2\text{,}$ then $x^2 = 4\text{.}$

Notice that both of these statements are true. But we need to go further to justify that all other numbers fail to make a true statement. For example, ``If $x = 5$, then $x^2 = 4$'' is false since $5^2 = 25$ and $25$ is not the same as $4$. But checking that one number isn't good enough because there are infinitely other numbers to check (including all decimals and fractions). We clearly can't check an infinite number of values, and so we need some sort of logical structure to explain how we can know that we've got all the solutions.

Notice in the naming of the set, we used the symbols $\{$ and $\}\text{.}$ These are known as set brackets. You can think of the set brackets as a bag in which you're going to put various objects. The contents in between the set brackets describe the objects in the set in some form. In our example, we presented that as a list. We would read $\{ -2, 2\}$ as "the set containing the numbers -2 and 2." We don't really care about the order that we list the values, so that this set is the same as $\{ 2, -2 \}\text{.}$

Some books go into some detail about set-builder notation, which is another way of describing sets. Rather than making an explicit list of elements, the set is defined by properties. Here is an example:

\begin{equation*} \{ x \in \mathbb{R} : x^2 = 4 \} \end{equation*}

You can immediately see that set-builder notation has a lot of symbols. We'll explain how to read this and then break it down into components.

\begin{equation*} \begin{array}{ccccc} \{ \amp x \in \mathbb{R} \amp : \amp \quad x^2 = 4 \quad \amp \quad \} \\ \downarrow \amp \downarrow \amp \downarrow \amp \downarrow \\ \quad \text{The set of} \quad \amp \quad \text{real numbers $x$} \quad \amp \quad \text{such that} \quad \amp \quad x^2 = 4 \quad \amp \end{array} \end{equation*}

We've already talked about the set brackets, so let's look at the other pieces

$x \in \mathbb{R}\text{:}$ This defines two objects. The first object it defines is a variable. This is a symbol that represents the quantity with the desired property. In this case it's the symbol $x\text{,}$ but it can be basically any symbol you want. The second object that this defines is the universal set. This is important because it defines the collection of objects that we're going to consider. In this case, we're thinking about real numbers.

The symbol $i$ is called the imaginary unit and represents the quantity $\sqrt{-1}\text{.}$
If you happen to be familiar with complex numbers, you will probably remember that there is a special number called $i$ that exist in the complex numbers that isn't part of the real numbers. For this situation, we're saying that we're not considering complex numbers to be options for our variable.
$x^2 = 4\text{:}$ This is the property that the variable needs to satisfy. In other words, in order to be in the set the variable must satisfy this equation. This part of the notation can be a single equation, multiple equations, or even a collection of words that describe the property.

This set is the solution set of the equation $x^2 = 4$ because it says that it is the set of all real numbers that make $x^2 = 4$ a true equation. And this shows one of the downsides of this notation. When we define sets in terms of properties, it may still leave us with work to do to figure out what values actually have the desired property. In order for us to know that this set contains the numbers -2 and 2, we would still have to do some work. And that's both the strength and weakness of set-builder notation.

On the one hand, the notation is flexible enough that we can defined objects by a list of properties. On the other hand, that flexibility sometimes means that even though we know what properties the objects should have, we often don't actually know what the exact objects are that are in the set.

Some books try to teach students set-builder notation and ask them to present their solutions to equations as sets. Unfortunately, this is makes things even more confusing and pushes students deeper into the world of rule following rather than understanding. This puts students either in the position where their notation is wrong, or their notation is correct but they don't really understand what they're writing. Neither is a good thing.

Set theory is the mathematical field that studies sets very carefully. It needs to be done carefully because as simple as it may appear here, it turns out to be an incredibly difficult subject. We won't be able to explore the details of that, but we can get a brief glimpse of some of the complexity that arises from that area.

Let's start with a set that contains nothing, which is knwon as the empty set. There are two notations for this: $\{ \}$ (literally, a set with nothing in it) and $\emptyset\text{.}$ The latter notation is far more common, so that's what we will use. We can think of this as being like an empty box. It's a collection of objects where there is nothing inside.

But now let's consider this object: $\{ \emptyset \}\text{.}$ This is the set that contains the empty set. This is mathematically different from the empty set in the same way that a box containing an empty box is not the same thing as an empty box. And you can already start to see some of the difficulty of this area of mathematics. You have to parse the symbols very carefully to understand what's happening, because even though the object inside the box is an empty box, we have to remember that the outside box is not actually empty.

Or consider this object: $\{ \emptyset, \{ \emptyset \} \}\text{.}$ This is the set that contains the empty set and the set that contains the empty set. In other words, it's a box with an empty box and a box containing and empty box. We might want to ask a basic question: How many objects are in this set? And this turns out to be somewhat challenging because there are actually four boxes in the picture now. What is the "correct" way of counting them?

While this may appear to be extremely abstract, this way of thinking turns out to be very closely related to how huge amounts of data is organized. For example, phones and computers use a file folder structure that looks like boxes inside of boxes, and the internet uses a lot of JSON data which can also be thought of as boxes inside of boxes. And so the abstract ideas of set theory end up having practical ways of being interpreted and applied.