*This post was originally published on September 11, 2018, and updated most recently on July 26, 2020.*

When looking back at the Venn diagrams you created in grade school, you probably have fond memories of charting which types of candy bars you and your friends liked or comparing your favorite movie characters. While you may have thought your Venn diagramming days were long behind you, these tools are actually useful throughout adulthood. In fact, mathematicians and related professionals use them to represent complex relationships and solve mathematical problems all the time.

Of course, the objects being studied in professional diagrams are usually not candy bars or movie characters. And there’s a lot more you need to understand to use them effectively. To fully embrace the world of professional Venn diagrams, you should have a basic understanding of the branch of mathematical logic called ‘set theory’ and its associated symbols and notation.

Using set theory, researchers and mathematicians have established the foundations of many mathematical concepts including diverse sets of structures, relations, and theorems that can be applied to various areas of study including topology, abstract algebra, and discrete mathematics.

Using the basics we’ll cover here, you too can begin using Venn diagrams in more complex ways.

## Venn diagram symbols

While there are more than 30 symbols used in set theory, you don’t need to memorize them all to get started. In fact, the following three are the perfect foundation.

### The union symbol ∪

Venn diagrams are comprised of a series of overlapping circles, each circle representing a category. To represent the union of two sets, we use the ∪ symbol — not to be confused with the letter ‘u.’

In the below example, we have circle A in green and circle B in purple. This diagram represents the union of A and B which we notate as A ∪ B.

Let’s revisit those grade school days for a moment with that example of candy bars. If circle A were people who like Snickers bars and circle B were people who like 3 Musketeers bars, A ∪ B would represent people who like Snickers, 3 Musketeers, or both.

### The intersection symbol ∩

The area where two sets intersect is where objects share both categories. In our example diagram, the teal area (where green and purple overlap) represents the intersection of A and B which we notate as A ∩ B.

This intersection is the area where we would find people who like both Snickers and 3 Musketeers.

### The complement symbol A^{c}

The categories not represented in a set are known as the complement of a set. To represent the complement of set A, we use the A^{c }symbol.

To represent the absolute complement of a set, i.e., everything not included in the set, we use the equation A^{c} = U \ A where the letter “U” represents the given universe. This equation means that everything in the universe, except for A, is the absolute complement of A in U.

The gray section of our example diagram represents everything outside A.

Using our candy bar example, this would represent everyone who does not like Snickers.

## Another example

Let’s try a new example. Say we’re planning a party at work, and we’re trying to figure out what kind of drinks to serve. We ask three people what drinks they like. When we ask, this is what we get:

Drink | A | B | C |

Wine | X | X | X |

Beer | X | X | |

Martini | X | X | |

Old Fashioned | X | X | |

Rum & Coke | |||

Gin & Tonic |

Using a three-circle Venn diagram, we can cover every possibility. Each person is represented by a circle, symbolizing them with A, B, and C. Using the ∩ symbol, we can show where intersections between sets should be placed.

When we fill in the diagram with our data, we place each object according to the formulas we indicated above. For example, we place the Martini in the B ∩ C area because respondents B and C indicated they enjoy them. Because Rum & Coke and Gin & Tonics weren’t selected by anyone, they do not go within any circle. However, because they still exist and are available in the universe, they can be placed in the white space.

Here’s our final diagram:

Clearly, wine is the best choice for our upcoming party. Beer, martinis, and old fashioneds may be good secondary drinks to offer, but they probably shouldn’t serve rum & cokes or gin & tonics.

## Venn diagram examples

Taking all of these versions with the symbology you’ve learned should serve as a great start to making the Venn diagrams that’ll help your team. Use the series of Venn diagram templates on Cacoo as a jumping-off point.

Here are a few more examples as you continue:

### How to read a Venn diagram

Now that you know all about how to make a Venn diagram and include the official terminology and symbology, you should understand how to correctly read one.

By reverse engineering, you can take the information already in the diagram to see where the symbols and equations we’ve laid out would sit. No matter how many options are added, you know where similarities or preferences lie for them, and the differences between what elements end up sitting inside and outside of the chart.

## Set theory

While we could go extremely in-depth on set theory (there’s always more to learn), a fitting way to finish off a lesson on Venn diagrams is by learning some of the theory behind them.

A **set** is a group or collection of things, also known as elements. Those elements could really be anything. From the example above, the set is the choices the unnamed group make for their drink preferences.

In set theory, we would write this out instead as an equation, listing all of the elements within curly brackets:

{person 1, person 2, person 3, person 4, …}

Since the question of the example is which drink their preference is, these people end up divided into groups by their choices:

Old fashioned = {X people}

Martini = {X people}

Beer = {X people}

Rum & coke = {X people}

Gin & tonic = {X people}

Because we are offering five different drink options, we end up with five individual sets, which are then represented within the Venn diagram.

## Final thoughts

We’ve stuck to basic examples here for the sake of clarity, but there’s a lot more information out there you can use to go more in-depth with set theory. In fact, Stanford’s encyclopedia entry on set theory is a great place to start.

As you explore more set relationships, visualizing your work with Venn diagrams is a powerful and easy way to communicate these relationships with ease.

When you’re ready to start creating your own Venn diagrams, look no further than our cloud-based diagramming tool Cacoo. Our library of shapes can help you easily create diagrams from scratch or you can start from one of our hundreds of pre-made templates to simply plug in your info and go.