Venn Diagrams

MathsDirect

Venn Diagrams are a way of representing sets visually.

A Venn diagram consists of (usually interlocking) circles contained within a rectangle. Each circle represents a set, with the rectangle representing the Universal Set.

For example, if we were considering the Natural Numbers less than 10 and wanted to look at the set

A = {odd numbers}

then we would have

When your Venn Diagram contains more than one circle, you need to be able to describe, using the correct notation, the connection between the sets.

In the diagram below	E = {Natural Numbers from 1 - 10}
	A = {multiples of 2}
	B = {multiples of 3}

There are two terms to be defined in this example.

Notice that 6 sits in the area where the two sets cross, because it is both a multiple of 2 and 3. This area is called the intersection of the sets A and B. We write this

If we want to talk about the numbers that are in either A or B (or both) we write this

This is called the Union of A and B.

Calculating the Union

We can write the number of members of a set as n(A).

The number of members of the intersection will therefore be written as

To calculate the number of members of the union, we can use the formula

The reason for subtracting the intersection, is that when we add the two sets together, the intersection is counted twice. This formula becomes important when we start calculating probabilities.

To see a trivial example of this formula; a group of students were surveyed on their A-levels. 20 were studying Mathematics and 15 were studying Physics. If 8 were studying both, how many students were surveyed?

The Complement

One final piece of notation that you should be aware of is the symbol that an element is not in a set.

The set of students not taking Maths is M'.

and

Finally, it is worth noting that in this case

That is

is a null set

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