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What are the Union and Intersection of Sets?

Easy ways to find the union and intersection of sets.

By
Jason Marshall, PhD,
July 8, 2011
Episode #069

At the beginning of the last article about sets and subsets, we talked about the relationship between the words blue, green, lilac, and rose. In particular, we noted that while these four words are all names of colors, the last two words—lilac and rose—are also names of flowers. This flowery example perfectly illustrates the concepts of mathematical sets and subsets. And now that we’ve got that down, we’re ready to see what this all has to do with the union and intersection of sets too.

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Review: What are Sets and Subsets?

A set in math is simply a group of things. For example, the set of all even positive integers less than 10 is { 2, 4, 6, 8 }, the set of all the numbers that are solutions to the two problems 100 / 25 and 30 + 5 is { 4, 35 }, and the set containing the names of the 50 United States is { Alabama, Alaska, Arizona, Arkansas, California, Colorado, …, Wisconsin, Wyoming }.

So, what can we do with sets? Well, one thing we can do is create new sets from them that contain subsets of their original elements. For example, we can make a new set from the set of United States by taking a subset of the states whose names begin with the letter “A.” This set has four elements: { Alabama, Alaska, Arizona, Arkansas }.

What Is the Union of Sets?

But that’s not all we can do with sets. Of particular interest for today are two specific operations—the first of which is called finding the “union” of sets. Imagine you have two sets: the set of the names of all the types of fruit sold at your local farmers’ market, and the set containing the names of your favorite fruit. Just for the sake of our example, let’s imagine that the first set—the set of all the fruit at your local farmers’ market—has five elements: { peach, nectarine, plum, apple, strawberry }. And let’s imagine that your favorite fruits form the three-element set:  { cherry, banana, peach }.

The union of these sets is the new set formed by combining the different unique elements from each of the individual sets. In other words, it’s the analog of addition for sets. In the case of our example, the union is the new set: { peach, nectarine, plum, apple, strawberry, cherry, banana }. Notice that the set of fruits at your farmers’ market contains five elements, the set of your favorite fruits contains three elements, and the union of these two sets contains seven elements. So we’ve added a set with five elements to a set with three elements to get a set with seven elements…which is one less than 3 + 5 = 8. What happened to that other one? Well, as we learned in the last article, a set cannot contain two copies of the same element. In this case, since peaches are sold at the farmers’ market and are also one of your favorite fruits, the union of the two sets contains only one of these two copies of the element “peach.”

What Is the Intersection of Sets?

If you think about it for a minute, you’ll see that there’s something else we can do to “add” the two sets in our fruit example. As well as “adding” the sets by finding the new set that contains all the unique elements in either set—that is, finding the union—we can also perform a type of addition by creating a new set from the elements that each of the two sets have in common. This process is called, quite logically, finding the intersection of the sets.

For example, the intersection of the set of all fruit at your local farmers’ market—{ peach, nectarine, plum, apple, strawberry }—and the set of your favorite fruits—{ cherry, banana, peach }—is a set that contains only a single element: { peach }. Why? Because “peach” is the only element that appears in both sets.

What Does this Have to Do With Numbers?

[[ AdMiddle2 ]]Now that we understand sets and subsets and unions and intersections, we’re ready to start putting the pieces together to see not only what this all has to do with fruit, but with numbers and math too. In particular, we’ve talked in the past about different types of numbers: positive integers (and zero), negative integers, rational numbers, and irrational numbers. As you’ll now recognize, each of these types of numbers actually forms a set, and each type also forms a subset of an even larger group of numbers. We’ll talk about exactly what that larger group is next week and how this is all related to finding the union and intersection of sets. And then we’ll continue on after that in the following weeks to talk about some more ideas in math that are based upon these key concepts.

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. This week’s number is 12,000 kilometers. What’s that? Well, did you hear about the asteroid that flew by Earth on June 27? That asteroid didn’t just fly by Earth, it flew by Earth really closely. In fact, it came within about 12,000 km of Earth—that’s closer than the orbit of some satellites! Thankfully though, it missed us. Although, in truth, this asteroid was pretty small—only about 20 meters in diameter—so it didn’t really pose a huge threat. In contrast, the asteroid that led to the demise of the dinosaurs 65 million years ago was about 10 km across—that’s 500 times bigger!

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 can be hard. In fact, many 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 in particular algebra—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. 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 each and 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!
 

 

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