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# What are Sets and Subsets?

How to use sets and subsets to understand the relationships between numbers.

By
Jason Marshall, PhD
Episode #68

What do the words “blue,” “green,” “lilac,” and “rose” have to do with each other? Well, for one thing, they’re all words that name colors. But that’s not all they are because the last two words—lilac and rose—are also used to name flowers – which means that there must be a relationship between words that name colors and words that name flowers. While that might not shock you, it turns out that this isn’t just any old relationship…it’s actually based upon one of the most important relationships in math. To help us understand this, today we’re going to take a look at the key mathematical concepts of sets and subsets.

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## What is a Set in Math?

So, what is a set? Well, a set in math is exactly what you’d expect: It’s simply a group of distinct objects. These objects can be mathematical things like numbers and geometric shapes, or they can be ordinary everyday things like a set of baseball cards or a set of silverware. The important thing is that these are all groups of distinct objects that when taken as a whole make up some other distinct object. For example, a spoon is a distinct object, but so is the set of silverware that’s made from a bunch of spoons, forks, knives, and so on.

A set can contain pretty much anything...even other sets! Actually, a set can even contain nothing at all—this special set is called the “empty set.” The individual objects in a set are called the elements of the set. So each of the individual spoons in a set of silverware is an element of that set. Or, as a more mathematical example: Each of the positive integers—1, 2, 3, and so on—is an element of the set of all positive integers. Simple, right? Well, believe it or not, this simple idea turns out to be critically important in math. In fact, it can actually be used as the cornerstone from which almost everything else can be derived! But that’s a much more advanced topic that’ll have to wait for another day.

## How Are Sets Defined and Written?

When you’re reading a math book (your favorite hobby, I know), sets are typically defined and written using pairs of curly braces to enclose all their elements. To make it easier to talk about and compare them to one another, sets are usually labeled with a capital letter. So, in a book, you might see something like U = {1, 2, 3} which means that there’s a set named “U” which contains three elements: the integers 1, 2, and 3. It’s also possible to define a set by describing it in words. For example, something like “W is the set of the colors of all the houses on your block” (which is a set with a finite number of elements) or perhaps “Y is the set of all the numbers in the Fibonacci sequence” (which is a set with an infinite number of elements).

The order that elements are written inside the braces describing a set doesn’t matter. If you have an apple, a banana, and an orange in a bag (or a set), it doesn’t matter how they’re arranged—it just matters that they’re in the bag. In other words, a set is defined by the distinct elements it contains. If you think about it, you’ll see that this means that a mathematical set cannot contain more than one of the same element. So, if you have two apples, a banana, and an orange in your bag of fruit, the set of fruit is still = { apple, banana, orange }.

A set is defined by the distinct elements it contains.

## What is a Subset in Math?

In next week’s article, we’re going to talk about some operations that can be used to construct new sets from existing ones. But to properly understand this, we first need to cover one more thing: the subset. So, what is it? Well, a subset is just a set that is composed entirely of members of some other set. For example, let’s say we’re talking about the set of all letters in the English alphabet—A, B, C, D, and so on through Z. One subset of this is the set of vowels—A, E, I, O, U. Another is the set of consonants—A, B, C, D, F, and so on. And a third is the set of letters used in the spelling of your name—J, A, S, O, N in my case. Of course, there are a bunch of other possible subsets too.

[[AdMiddle]There’s a lot more we can do with sets and subsets, but we’re going to leave things right there for now and pick up the topic again next time. In the meantime, take a few minutes to think about some things—they can be mathematical things or real objects in the real world—whose relationships can be described in terms of sets and subsets. Just look around you and you’ll see that the world is full of them!

## Number of the Week

Before we finish up today, it’s time for this week’s featured number selected from the various numbers of the day posted to the Math Dude’s Facebook page. I couldn’t decide on just one number, so this week we have two: 75 and 200 miles per hour (mph). What are they? They’re the speeds of the fastest animals on land and in the air. That’s right: a cheetah can sprint at up to 75 mph and a peregrine falcon can dive while hunting at 200 mph! Pretty amazing speeds, right?

Of course, it helps that the peregrine falcon has gravity on its side. After all, when a cheetah gets moving at 75 mph, its muscles have to do all the work. But when a falcon drops in at 200 mph, it's sort of just letting gravity do its thing. Don't get me wrong, it's still highly impressive to be able to fly, catch prey, and come out of a 200 mph dive safely; but if you drop me or a cheetah from a plane, and we get in good aerodynamic positions, we can probably get close to 200 mph too! Although we probably won’t be able to catch prey…or land without breaking a few bones.

## New Math Dude Algebra Book!

Exciting news: My new book, The Math Dude’s Quick and Dirty Guide to Algebra, is now available! You can get your copy from Amazon, Barnes & Noble, Powell’s, the iBookstore, or your favorite bookstore.

What’s it about? Well, as I’m sure you know, algebra is hard for a lot of people. In fact, a lot of people aren’t even sure what algebra is! But things don’t have to be that way. In this book, I invite you to check your confusion at the door and enter a new world in which math—and algebra, in particular—actually makes sense. Using detailed explanations, lots of brain teaser puzzles, and even secret-agent “math-libs,” I’ll take you step-by-step through learning and truly understanding the most important parts of algebra so that you can get rid of that “I have no idea what any of this means” feeling forever. In other words, you’re just one step away from finally making sense of it all, so do yourself a favor and pick up a copy of The Math Dude’s Quick and Dirty Guide to Algebra today. Thanks for checking it out!

## Wrap Up

Okay, that’s all for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new number of the day and math puzzle posted every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.com.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

## About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.