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.

A Venn diagram for Hockey and gymnastics. The circle for Hockey is shaded. Ask your teacher for more information.

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 students play hockey.

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

A Venn diagram for Hockey and gymnastics. The area outside the circles is shaded. Ask your teacher for more information.

The number outside box are the students who do not fit into either category.

This means there are 15 students who do not 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.

A Venn diagram for 2 sets Hockey and gymnastics. The intersection is shaded. Ask your teacher for more information.

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

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

A Venn diagram for Hockey and gymnastics. Hockey is shaded but not the intersection. Ask your teacher for more information.

The part of the circle highlighted here is all the students who play hockey but do not do gymnastics. We could refer to these as "the students who only play hockey".

A Venn diagram with 2 sets that do not overlap.

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.

A Venn diagram with 2 sets where one set is completely inside the other set.

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

Examples

Example 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".

A Venn diagram with 2 sets: not hungry and too busy that overlap. Ask your teacher for more information.

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

Worked Solution

Create a strategy

Find the "Not hungry" circle in the diagram and add the number of students inside it.

Apply the idea

\displaystyle \text{Not hungry students}	\displaystyle =	\displaystyle 8 + 11	Add numbers in the not hungry circle
	\displaystyle =	\displaystyle 19	Evaluate

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

Worked Solution

Create a strategy

Find the number that is only in the too busy circle.

Apply the idea

13 is in the "Too busy" circle and not in the "Not hungry" circle.

\text{Too busy students}=13

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

Worked Solution

Create a strategy

Add the number of students that are in the "Not hungry" circle or in the "Too busy" circle but not in both.

A Venn diagram with 2 sets that overlap. The circles are shaded but not the intersection.

Apply the idea

\displaystyle \text{Students with one reason}	\displaystyle =	\displaystyle 8 + 13	Add the numbers
	\displaystyle =	\displaystyle 21	Evaluate

Idea summary

Venn diagrams are a type of diagram that uses circles to group and organise things.

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.

A Venn Diagram with 3 overlapping circles labeled Comedy, Action, and Horror.

Here is a diagram representing three movie genres.

A Venn diagram with 3 overlapping circles. The upper left circle is shaded.

This highlighted circle represents comedy movies.

A Venn diagram with 3 sets overlapping. The shaded part is the intersection of the 3 sets.

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

A Venn diagram with 3 sets overlapping. The intersection of 2 sets is shaded. Ask your teacher for more information.

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.

Examples

Example 2

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.

A Venn diagram with 3 sets overlaps: comedy, action, and horror. Ask your teacher for more information.

How many of the movies are horror films?

Worked Solution

Create a strategy

Add all the numbers inside the "Horror" circle.

Apply the idea

\displaystyle \text{Horror films}	\displaystyle =	\displaystyle 13 + 3 + 5 + 6	Add the numbers inside horror
	\displaystyle =	\displaystyle 27	Evaluate

How many of the movies fit into only one genre?

Worked Solution

Create a strategy

Add the numbers in these parts where the movies fit only one genre:

A Venn diagram with 3 sets overlapping. All sets are shaded excluded the intersection. Ask your teacher for more information.

Apply the idea

\displaystyle \text{Number of movies}	\displaystyle =	\displaystyle 10+20+13	Add the numbers
	\displaystyle =	\displaystyle 43	Evaluate

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

Worked Solution

Create a strategy

Add the numbers in these parts where the movies are action combined with at least one other genre:

A Venn diagram with 3 sets overlapping. Ask your teacher for more information.

Apply the idea

\displaystyle \text{Number of movies}	\displaystyle =	\displaystyle 4 + 5 + 6	Add the numbers
	\displaystyle =	\displaystyle 15	Evaluate

Idea summary

Three circle Venn diagram is a diagram shows how components of three sets are related using three overlapping circles.

17.06 Venn diagrams

Ideas

Venn diagrams

Examples

Example 1

Three circle venn diagrams

Examples

Example 2

Outcomes

MA4-21SP

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