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 it’s 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
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 Ac
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 Ac symbol.
To represent the absolute complement of a set, i.e., everything not included in the set, we use the equation Ac = 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.
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:
|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.
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.