Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Some people like to say that math made sense when it was all numbers, but then things fell apart when the letters started showing up. This is unfortunate, because the letters are actually just a way of representing numbers, and thinking about them in that way can help to clarify confusion. There are a lot of ways that teachers talk about variables and they are of varying degrees of correctness. For example, some teachers say that variables represent unknown values. It’s not exactly wrong, but it’s not exactly right. Whether or not we know the value is irrelevant. In college level math courses, we can use variables to represent mathematical expressions with other variables in it instead of just thinking of it as a quantity. This leads us to a more general definition:

Notice that we allow for variables to represent both quantities and mathematical expressions. What this means is that both \(x = 4\) and \(x = y + 1\) are valid uses of variables.

PDF Version of these Worksheets

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external/worksheets/02-Worksheets.pdf

Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.

Most textbooks will have hundreds of problems for you to work on, especially for a section like this one. And there’s certainly value to practicing those manipulations over and over again so that you can get a lot of experience. However, the approach of this textbook is much more about slowing you down to help you create a solid mental framework for understanding mathematical ideas.

One piece of that framework of mathematical thinking is the ability to take something complicated and break it down into simpler pieces. One of the ways that manifests is the transition from working with numbers to working with variables. We tried to set you up with this idea in the forward direction by giving you a series of equations with numbers before asking you to work with variables:

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.

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 \(\{ 2, -2 \}\text{.}\)

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.

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

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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.