Venn Diagrams (Edexcel GCSE Maths: Foundation)

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Mark

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Maths

Set Notation

What is a set?

A set is a collection of elements
- Elements could be anything
  - Numbers, letters, coordinates, ...
You could describe a set by writing its elements inside curly brackets {}
- {1, 2, 3, 6} is the set of factors of 6
If the set of elements follow a rule then you can write this using a colon inside the curly brackets {... : ...}
- The bit before the colon is the type of element
- The bit after the colon is the rule
  - {x is a positive integer : x² < 30} is the set of positive integers which, when squared, are less than 30
  - This is equal to {1, 2, 3, 4, 5}
- The colon is often read as 'such that'
- If no type is specified, x can take any value (fractions, decimals, irrationals, ...)
  - {x: x² < 30} means any value whose square is less than 30

What do I need to know about set notation?

$E$ is the universal set (the set of everything)
- For example, if we are only interested in factors of 24 then $E$ = {1, 2, 3, 4, 6, 8, 12, 24}
- You may see alternative notations used for $E$
  - U is a common alternative
  - S or the Greek letter ξ (xi) may also be seen
We use upper case letters to represent sets (A, B, C, ...) and lower case letters to represent elements (a, b, c, ...)
n(A) is the number of elements in set A
- For example, if $E$ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 4, 9}, B = {1, 2, 3, 4, 5, 6}
  n(A) = 3, n(B) = 6
∈ means "is an element of"
- e.g. 3 ∈ {1, 2, 3, 6} because 3 is an element of the set {1, 2, 3, 6}
A ∩ B means the intersection of A and B (the overlap of A and B)
- This is the set of elements that are in both set A and set B
  A ∩ B = {1, 4}
A U B means the union of A and B (everything in A or B or both)
- This is the set of elements that are in at least one of the sets
- This includes elements in both sets (in the intersection)
  A U B = {1, 2, 3, 4, 5, 6, 9}
A' means the complement of A
- It is the set of all elements in the universal set $E$ that are not in A
  A' = {7, 8, 9, 10}

Sets & Venn Diagrams

What is a Venn diagram?

A Venn diagram is a way to illustrate all the elements within sets and any intersections
A Venn diagram consists of
- a rectangle representing the universal set ( $E$ )
- a circle for each set
  - Circles may or may not overlap depending on which elements are shared between sets

What do the different regions mean on a Venn diagram?

$A \cap B$ is represented by the region where the A and B circles overlap
$A \cup B$ is represented by the regions that are in A or B or both

Venn diagrams showing the union and the intersection of sets A and B

Worked example

Two sets A and B are shown in the Venn diagram.

Two sets, A and B, represented on a Venn diagram

(a)

Write down n(A).

The elements of A are anything inside the A circle

A = {2, 6, 12, 14, 28}

n(A) means the number of elements in A
There are 5 elements in A

n(A) = 5

(b)

Use set notation to complete the sentence {14, 28} = ...

14 and 28 are the elements that are both in A and B
This means they are in the intersection of A and B

{14, 28} = A ∩ B

(c)

Write down the elements that are in set A U B.

A = {2, 6, 12, 14, 28} and B = {7, 14, 21, 28, 35}
A U B is the set of elements that are in at least one of the sets
For elements in both, only write them out once

A U B = {2, 6, 7, 12, 14, 21, 28, 35}

Probability & Venn Diagrams

How do I find probabilities from Venn diagrams?

Count the number of elements you want and divide by the total number of elements
For the Venn diagram shown below,
- The probability of being in A is
  - There are 5 elements in A out of 11 in total
- The probability of being in both A and B is
  - There are 2 elements in A and B (the intersection)
- The probability of being in A, but not B, is
  - 3 elements are in A but not B
Some harder questions are not out of the total number, but out of a restricted number
- The probability of being in B, given that you are already in A, is
  - You are only interested in elements in A
  - There are 5 elements in A, out of which only 2 are also in B

Two sets, A and B, represented on a Venn diagram

Exam Tip

Be careful when filling in numbers for a Venn diagram
- Some of the given numbers may need to be split between two sections of the Venn diagram
Suppose 10 people have a cat, 8 people have a dog and 6 people have both a cat and a dog
- Out of the 10 people who have a cat
  - 6 also have a dog
  - 4 do not have a dog
- Out of the 8 people who have a dog
  - 6 also have a cat
  - 2 do not have a cat

Worked example

In a class of 30 students, 15 students study Spanish and 3 of the Spanish students also study German.
7 students study neither Spanish nor German.

(a)

Draw a Venn diagram to show this information.

Draw the Venn diagram with its rectangular box and two (labelled) overlapping circles
3 students study both Spanish and German, so start here and work outwards
12 must study Spanish but not German (to get 15 in total for Spanish)
7 study neither, so this goes outside of the circles
To get 30 in total, 8 must study German but not Spanish

Venn diagram showing the numbers of students studying Spanish and German

(b)

Use your Venn diagram to find the probability that a student, selected at random from the class, studies Spanish but not German.

It helps to highlight Spanish but not German

Venn diagram showing the numbers of students studying Spanish and German with the the number studying Spanish only, highlighted

Divide the number of students studying Spanish but not German by the total number of students

Students studying Spanish but not German = 12
Total number of students = 30

P(Spanish but not German) $= \frac{12}{30} (= \frac{2}{5})$

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