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# What is a Venn Diagram?

Learn how to make and use Venn diagrams to help with math, English, and many other subjects.

By
Jason Marshall, PhD,
August 5, 2011
Episode #072

Some things in math have gone mainstream. Arithmetic is used by pretty much everybody on a daily basis and lots of other ideas from math are used outside the cozy world of the classroom, too. For example, I have it on good authority (from my wife) that today’s math topic is frequently used in high school English lessons. What could it be? Well, today we’re talking about Venn diagrams.

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## How to Make a Venn Diagram

You probably remember having to draw Venn diagrams in school. They’re fairly simple to make and also fairly simple to understand. But don’t let that simplicity fool you—Venn diagrams actually incorporate all of the important ideas about sets and subsets and unions and intersections that we’ve talked about in recent articles.

To make a Venn diagram, start by drawing a rectangle. The area inside this rectangle represents all the possible things contained in our diagram. In our case, let’s say that the area inside our rectangle represents the set of all positive integers: { 1, 2, 3, 4, ... }.

The next step is to draw overlapping circles inside this rectangle. The area inside each of these circles represents a subset of the things within the big rectangle. So, in this example a circle represents a subset of the set of positive integers. Let’s start by drawing a circle containing the positive integers that are greater than 5—that’s the set { 6, 7, 8, 9, 10, … }. And then let’s draw another circle (that overlaps the first) containing the positive integers that are less than 10—that’s the set { 1, 2, 3, 4, …, 8, 9 }.

## What Do Overlapping Regions in Venn Diagrams Mean?

Why do these circles have to overlap? Well, the purpose of a Venn diagram is to show all the possible relationships between the various sets represented by each of the circles. And, as we’ll see in a minute, all of these relationships are described by the various overlapping regions. In our case, the two circles representing positive integers that are greater than 5 and positive integers that are less than 10 have an overlapping region that represents the set of positive integers that are both greater than 5 and less than 10—in other words, the set { 6, 7, 8, 9 }.

Now let’s add another circle to our diagram that represents all even positive integers—that is, all positive integers that are divisible by 2. This new circle must overlap each of the previous two that we’ve already drawn. If you draw this, you’ll see that there are a total of seven regions within the circles of this Venn diagram: Three representing the big sets contained in the original circles; three representing the various combinations of pairs of these sets: that’s the simultaneously greater than 5 and less than 10 region, the simultaneously greater than 5 and even region, and the simultaneously less than 10 and even region; and, last but not least, there’s a region in the center of our Venn diagram representing positive integers that are simultaneously greater than 5, less than 10, and even—which is the pair of numbers { 6, 8 }.

## Unions, Intersections, and Venn Diagrams

[[AdMiddle]Does anything about this idea of overlapping circles in Venn diagrams seem familiar to you? Perhaps it reminds you of the ideas of unions and intersections of sets? Indeed, if you think about it, you’ll see that the Venn diagram we’ve drawn is just a fancy way to visualize the properties of sets. You can find the intersection of two or more sets using a Venn diagram by finding the region where they overlap. For example, the overlapping region between the sets of numbers greater than 5 and less than 10 is just the intersection of these two sets. You can also use Venn diagrams to find the union of two or more sets. For example, the union of the set of even numbers and the set of numbers greater than 10 is the region of our Venn diagram contained within either of the two circles that represent these sets of numbers.

## What Can Venn Diagram Represent?

But numbers aren’t the only things that circles in Venn diagrams can represent. In fact, Venn diagrams can be used to show pretty much anything. There’s a famous one that circulated on the Internet a while ago that explains the difference between “geeks,” “dweebs,” “dorks,” and “nerds.” In that case, each circle represents a different set of human characteristics. Venn diagrams are also commonly used to represent and compare things like answers to surveys, possible hands in card games, and character traits and themes in novels (that’s the English class tie-in mentioned earlier).

## Number of the Week

Before we finish up, 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 10.7. What’s that? It’s the height of Mt. Everest—the tallest mountain on Earth—written in terms of the height of the tallest building on Earth—the 160-floor (2,717-foot) Burj Khalifa in Dubai. If you stack up 11 of these buildings one on top of the other, they’d reach the top of Mt. Everest. Imagine climbing those stairs!

## How Did Last Week’s Number Trick Work?

One last thing: Did you figure out how the second number trick we talked about last week works?

The trick went something like this: Think of a number between 1 and 10. Now, double that number and then add 10 to the result. Next, divide this new number by 2, and then subtract your original number. When you do that, you’ll always get 5. But why?

Let’s use the question mark symbol, “?”, to represent the number we pick between 1 and 10. When we double this number, we get a value of 2 x “?”. Then, if we add 10 to it we get: 2 x “?” + 10. Dividing the whole thing by 2 (which we can do by multiplying by the fraction 1/2 and using the distributive property) gives us “?” + 5. Finally, when we subtract our original number, “?”, we’re left with “?” + 5 – “?” = 5. The answer always has to be 5 because this whole thing was really just an algebraic equation…one that you can use to impress your friends!

## 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!