Venn Diagrams

Keeping track of relationships can be difficult if you try to do it all in your head, but fortunately, you don't have to. A man named John Venn invented a visual aid to help with these relationships back in 1880. Today, these Venn diagrams bear his name and help students all over the world remember everything from mathematical relationships to the motivations of characters in a Shakespearean play.

"How do Venn diagrams work?" you might be wondering. The answer is a series of intersecting circles where everything that fits into multiple groups appears in the intersection while anything that fits into one group appears outside of the intersection. If anything doesn't fit into any of the categories, it appears outside of all of the circles. Let's take a look at an example:

In this example, $R$ represents all reptiles, $S$ represents all animals that live in the sea, and $M$ represents all mammals. The funny-looking $\cap$ symbol means "intersection" and can be used to express where these categories intersect. For example, $R\cap S$ represents the intersection between reptiles and animals that live in the sea, so any reptiles that live in the sea go there. There is no $R\cap M$ because reptiles and mammals are mutually exclusive, so their circles don't touch in the Venn diagram above.

Do Venn diagrams have to be so abstract?

Many Venn diagrams have specific examples written into the circles, so here is an example of what that looks like using the same categories from above:

The crocodile and lizard are reptiles that don't live in the sea, so they're in the green part previously labeled $R$ . The sea turtle is a reptile that lives in the sea, so it goes in the pale yellow section represented by $R\cap S$ . Starfish and tuna are sea creatures that aren't reptiles or mammals, so they're in the yellow section $S$ . The mammals (blue whale, armadillo, and horse) follow the same rules, while the ostrich and cockroach don't fit any of the categories and therefore do not appear in any circles.

Do Venn diagrams have to be so fancy?

Absolutely not! Most students simply draw two or more interlocking circles on a piece of scrap paper and fill in the examples they need for the issue at hand. The graphic above probably looks nicer than anything you would draw, but all that matters is that it gets the job done.

Similarly, you might be surprised by how often a Venn diagram can help. The example above is rooted in science, but you might also see them in your English or social studies classes to help you understand relationships between people, characters, or countries. Of course, they come up in math too.

Venn diagram practice questions

Use the following Venn diagram to answer the questions:

a. What numbers are included in circle A?

$\left\{1,2,3,4,5\right\}$

b. What numbers are included in $B\cap C$ ?

$\left\{3\right\}$

c. How would you express where 5 is located above?

$A\cap D$

Topics related to the Venn Diagrams

Operations on Sets

Sets

Subsets

Flashcards covering the Venn Diagrams

8th Grade Math Flashcards

Common Core: 8th Grade Math Flashcards

Practice tests covering the Venn Diagrams

MAP 8th Grade Math Practice Tests

8th Grade Math Practice Tests

Get help with Venn diagrams today with Varsity Tutors

Venn diagrams are an invaluable study tool for students of all ages and ability levels, so your student might not be prepared for the rigors of more advanced coursework if they don't understand how to utilize them. Luckily, the Educational Directors at Varsity Tutors would be happy to explain how private tutoring can help students reach for their full educational potential. Contact us right now to get started!