Understand Venn Diagrams and Use Set Notation

Do you play any sports?

 

 

Year 9 students at the Edplace school play either rugby or football (or both or neither).

 

How many play each is represented in the following Venn diagram:

 

 

We can see that 8 students play both football and rugby as there is 8 where the two circles overlap, i.e. the intersection.

 

If we call the students who play rugby set R and those who play football set F, then their intersection is denoted by R ∩ F.

 

So R ∩ F = 8

 

If we knew that these 8 students are Amirah, Juan, Fatima, Alex, Emma, Mei, Tyrone and Hannah, then we can write:

R ∩ F = {Amirah, Juan, Fatima, Alex, Emma, Mei, Tyrone, Hannah}

 

Curly brackets are used if we want to list the actual elements in the set (i.e. the individual students who play both sports here).

 

If there are no curly brackets and there is just a number after the equal sign (like in R ∩ F = 8), then the number is how many elements there are in the set.

 

 

We can read other things from the diagram, such as that 10 students play rugby only or that 5 students play neither football nor rugby.

 

 

What about the number of students who play rugby or football?

 

Well, that's all the numbers in the circles added together!

 

10 + 8 + 12 = 30 students play rugby or football.

 

We call this the union. It's denoted by ∪ so R ∪ F = 30

 

One thing to note here - in maths, when we say 'or', we don't mean 'either ... or ...'

Rather, we mean '... or ... or both'

That's why the union includes the 8 in the intersection!

 

Let's put all this into practice!