Venn Diagrams ( GCSE / IGCSE )

Learn how to complete a Venn diagram by watching how I fill in the numbers for a typical exam question involving the intersection and union of sets. #mathstutor #mathsteacher http://www.davidwright.uk

Transcript:

Hi everyone. This is the first video in a series I’m creating about Venn diagrams.

To get started, we’re going to look at a typical exam question you might get.

In this question, we’re told that there are one-hundred students in a group,

and we’re then given the following data about whether they own a mobile phone or computer.

The question asks us to use this information to complete the Venn diagram, where we need to add four numbers.

Venn diagrams are used to represent sets, and the first thing I want you to note is the symbol here, which means the universal set.

This is the set that contains all the elements which, in this question, is one-hundred.

So this means that the numbers we enter anywhere within this Venn diagram need to add up to one-hundred.

This oval represents to set of students that own a mobile phone, so the numbers entered in this oval need to add up to sixty-eight.

This oval represents the set of students that own a computer, so the numbers entered in this oval need to add up to seventy-eight.

And the number entered outside of the oval shapes, which can be placed anywhere in this rectangle, represents the students that do not own a mobile phone or a computer.

This is the easiest one to fill in, so we can write ten here because this is clear from the information provided.


The tricky number to find is the one here because this number represents the intersection between the mobile phone set and the computer set.

The number in this region represents how many students own a mobile phone AND a computer, but this is not given in the information provided so I’m going to show you how to work this out.

Well, to do this, the first thing to note is that the whole of this region represents the union of the sets and the numbers anywhere in this region need to add up to ninety, because we are told that the universal set is one-hundred, and the number outside of this region is ten, meaning that there are another ninety elements that need to be entered on the diagram.

So we need to find this number, and I’ve moved the important information to the top of the screen.

As already stated, we know that these three numbers add up to ninety.

But this number is unknown, so I am going to call it x.

This means that this number must be sixty-eight minus x, because sixty-eight students own a mobile phone and so all of the numbers in the whole of this oval need to add up to sixty-eight.

And this number must be seventy-eight minus x, because seventy-eight students own a computer, so these numbers need to add up to seventy-eight.

And because the union of these sets is ninety we can write the following equation, which uses algebra to show that there are three numbers we want to find and they add up to a total of ninety.

We can simplify this as follows, because sixty-eight plus seventy-eight is one-hundred and forty-six.

And also, minus x plus x cancels, leaving just one minus x.

If we subtract one-hundred and forty-six from both sides we get the following.

And if we multiply by minus one we find that x equals fifty-six.

So we have found the value of x.

And we now know that fifty-six students own both a computer AND a mobile phone which is represented by this overlapping intersection region.

To find the number of students that own a mobile phone only, we write sixty-eight minus fifty-six, which is twelve. This is because the numbers in the whole of this oval need to add up to sixty-eight.

And for the number of students that own a computer only, we have seventy-eight minus fifty-six, which is twenty-two because both of these numbers need to add up to seventy-eight.

So we have now found all of the numbers and have completed the Venn diagram.

It’s a good idea to check that these number add up to ninety, which they do, because then ninety plus ten is one-hundred which is the universal set.

So our Venn diagram is correct and we have learnt how to find the unknown number at the intersection.

It is my hope that this video as helped you.

Best of luck with your studies and don’t forget to subscribe if you want regular help with your maths.