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Middle Years

12.03 Venn diagrams


Venn diagrams

Venn diagrams are a type of diagram that uses circle to group and organise things, as seen below:

A Venn diagram is a helpful tool in displaying information about two categories, especially if things can be in both categories. Some Venn diagrams will use the diagram to sort the different objects. The one shown above uses numbers in each section to show how many objects fit into that section.


We can look at various parts of the a Venn diagram to understand it more. The circle on the left represents every student that plays hockey, so adding the numbers in the circle together tells us that $5+6=11$5+6=11 students play hockey.

Similarly the numbers in the right circle will tell us how many students do gymnastics, there are $6+3=9$6+3=9 of those.


The number outside box are the students who don't fit into either category:

This means there are $15$15 students who don't participate in either hockey or gymnastics.


Some Venn diagrams wont have the outside box. This usually means that everything fits into one of the categories, or that everything outside those categories is being ignored.

The very middle section highlighted below represents students that are in both categories:

This section represents the $6$6 students who play both hockey and gymnastics.


The part of the circle highlighted below is all the students who play hockey but don't do gymnastics. We could refer to these as "the students who only play hockey":


Did you know?
Venn diagrams were originally just the diagrams that look like the one we've just been exploring. There are other ways we could use circles to draw diagrams, like this:

The proper name for these are Euler diagrams, but most people use these two terms interchangeably these days.

Worked example

Volunteers trying some free food were given the choice of two sauces to have with their chicken nuggets, as shown in the Venn diagram below. How many people didn't try both sauces?

Think: The region that corresponds to people that tried both sauces is this one:

The question wants the people that didn't try both, so we are after the sum of the numbers in all the other regions:


Do: Adding the numbers in all of the regions we are interested in gives $7+3+2=12$7+3+2=12. This means there were $12$12 people that did not try both sauces.

Reflect: We could also think of these regions as the "people that only tried one sauce" together with "the people that tried no sauces".


Three circle Venn diagrams

Venn diagrams can be drawn with three circles. These use the same concepts as a two circle Venn diagram but with another circle, represeting a third category, drawn overlapping the other two. We can work things out in the same way, but there are many more places that the categories overlap. Here is a diagram representing three movie genres:

The highlighted circle below represents comedy movies:

There are also some new possibilities such as all three at once:


Or even exactly two:


This region represents comedy (it lies in the Comedy circle) and horror (it lies in the Horror circle), but not action (it lies outside the Action circle). Remember to always look at each circle one at a time.


Practice questions

Question 1

A group of students were asked why they skipped breakfast. The two reasons given were that they were "not hungry" and they were "too busy".

  1. How many of the students skipped breakfast because they were not hungry?

  2. How many of the students only skipped breakfast because they were too busy?

  3. How many of the students skipped breakfast because of one reason?

Question 2

A group of students were asked about their siblings. The two categories show if they have at least one brother, and if they have at least one sister.

  1. How many of the students have at least one sibling?

  2. How many of the students have at least one brother?

  3. How many of the students don't have a sister?

Question 3

Joanne is struggling to decide what to watch online. She decides to pick one movie at random from the streaming website. A Venn diagram of her options sorts movies into three categories based on their genre: Comedy, Action and Horror

  1. How many of the movies are horror films?

  2. How many of the movies fit into only one genre?

  3. How many of the movies are an action film combined with at least one other genre?

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