1. Dr J Frost (jfrost@tiffin.kingston.sch.uk)
GCSE Sets
Dr J Frost
@DrFrostMaths
Last modified: 18th September 2017

2. What is a set?
In maths, it’s often useful to represent a collection of items.
We use curly braces to indicate a set of items...
−4, 1, 3 !
A set is a collection of items with 2 properties:
a. It doesn’t contain duplicates. ?
b. The order of the elements does not matter (but we usually write the items in ascending order) ?

3. What is a set?
Is it a set?
−3.5, 2, 9
False
True
4, 5, 5, 6
False
True 1
A set with one thing in it is known as a singleton.
False
True
It’s possible to have a set of sets!
1,2 , 3,4
False
True
𝑟𝑒𝑑,𝑏𝑙𝑢𝑒,𝑔𝑟𝑒𝑒𝑛
False
True
Sets need not consist of numbers.
Are these sets the same?
False
True
3,1,2 ={1,2,3}

4. −4, 1, 3 Finite Sets vs Infinite Sets
The examples with seen have been finitely large sets.
−4, 1, 3
But it’s also possible to have sets which are infinitely large…
(At A Level and beyond, the symbol ℕ is used for such a set)
“the set of all positive integers (whole numbers)”
“the set of all odd numbers”
Far far beyond the syllabus:
We could construct such a set using
2𝑘+1 𝑘∈ℤ }
which means “all possible numbers of the form 2𝑘+1, where 𝑘 is any integer”

5. ∅ The Empty Set We can also have a set with nothing in it!
(think of it as an empty bag)
It is known as the empty set: ∅ ?
Fro Fact: This is (as far as I know) the only Scandinavian letter used in maths.
We’ll see why it’s useful next lesson.

6. Venn Diagrams
Venn Diagrams are a way of showing the items in each set.
What does this region represent?
The items in 𝑨 but not in 𝑩.
What does this region represent?
The items in 𝑨 and in 𝑩. ? ? 𝜉 𝐴 𝐵 6 2 1 8 3 10 −1 7
What does this region represent?
The items neither in 𝑨 nor 𝑩.
Why the rectangular box?
It represents the set of all items we’re interested in. We use the special symbol 𝝃 (Greek letter “xi”) ? ?

7. 𝜉 𝐴 𝐶 𝐵 Example 1 2 3 4 5 6 7 8 9 15 𝝃= whole numbers from 1 to 15
𝑨= set of all prime numbers
𝑩= set of all numbers one less than a power of 2
𝑪= set of all square numbers
(Click to move!) 1 2 3 4 5 6 𝐶 7 8 9 15 𝐵

8. 𝜉 𝐴 𝐶 𝐵 Test Your Understanding ? Venn Diagram 10 8 2 4 1 5 9 3 6 7 ?
𝝃= whole numbers from 1 to 10 𝜉
? Venn Diagram 𝐴 10
𝑨= set of all cube numbers 8 2
𝑩= set of all odd numbers 4
𝑪= set of all multiples of 3 1
Bonus: If we extended 𝜉 to include more positive integers, what’s the smallest number that would appear in all three of 𝐴,𝐵,𝐶? 27 5 9 3 6 7 𝐶 𝐵 ?

9. Further Examples 𝝃 𝝃 ? ? 𝑨 𝑨 𝑩 3 𝑩 3 1 2 4 2 5 6 6 1 4 5 1
𝜉= 1,2,3,4,5,6 𝐴= 2,3,4 𝐵= 4,5
Construct a Venn Diagram to show these sets. 2
𝜉= 1,2,3,4,5,6 𝐴= 1,2,3 𝐵= 1
Construct a Venn Diagram to show these sets. 𝝃 ? 𝝃 ? 𝑨 𝑨 𝑩 3 𝑩 3 1 2 4 2 5 6 6 1 4 5
Any number in 𝐵 is also in 𝐴. It would therefore be a good idea to draw 𝐵 inside 𝐴 to show this relationship.

10. ! Elements of Sets and Subsets True or false?
An ‘element’ or ‘member’ is an item in a set. !
2∈ 1,2 means that 2 is a member of the set {1, 2}, i.e. it belongs to it.
Fro Note: ∈ is a special form of the Greek letter epsilon. It just means “is a member of”.
3∉ 1,2 means that 3 is not a member of the set.
𝐴⊂𝐵 means that set 𝐴 “is a subset of” 𝐵 It means anything in 𝐴 must also be in 𝐵 (and 𝐴≠𝐵)
True or false?
4∈ 1,3,4,5
2∉𝑃
2,3,4 ⊂ 3,4,5
0,2,3 ⊂{−1,0,1,2,3,4}
False
True
(where 𝑃 is the set of all prime numbers)
False
True
False
True
False
True

11. Exercise 1 ? ? ? ? ? ? ? 1 4 𝝃 𝝃 𝑨 𝑩 4 𝑩 6 5 8 7 𝑨 2 5 𝝃 𝝃 𝑩 𝑨 𝑪 2 𝑨 7
𝜉= 4,5,6,7,8 , 𝐴= 5,6,7 𝐵= 6,7,8
Construct a Venn Diagram to show these sets.
You have two sets 𝐴 and 𝐵 and 𝐵⊂𝐴. Draw a Venn Diagram (without any numbers) that indicates the relationship between the sets. 4 𝝃 ? 𝝃 ? 𝑨 𝑩 4 𝑩 6 5 8 7 𝑨 2
𝜉= 1,2,3,4,5,6,7,8 𝐴= 1,2,3,8 , 𝐵={3,4,5,8} 𝐶= 1,5,6,8
Construct a Venn Diagram to show these sets. 5
You have three sets 𝐴, 𝐵 and 𝐶 and 𝐴⊂𝐶. Draw a Venn Diagram (without any numbers) that indicates the relationship between the sets. ? 𝝃 𝝃 ? 𝑩 𝑨 𝑪 2 𝑨 7 3 1 8 4 6 5
The power set of a set is the set of all possible subsets, including the empty set and itself. E.g.
𝑃 1,2 = ∅, 1 , 2 , 1,2
Determine 𝑃 1,2,3 = ∅, 𝟏 , 𝟐 , 𝟑 , 𝟏,𝟐 , 𝟏,𝟑 , 𝟐,𝟑 , 𝟏,𝟐,𝟑
Determine how many members 𝑃 𝐴 has for a set of 𝐴 of size 𝑛. For each possible subset, each of the 𝒏 members of the original set can either be included or not included. That’s 2 possibilities, so 𝟐 𝒏 possible subsets. 𝑩 N 𝑪
𝜉= 1,2,…,10 𝐴= set of all primes
𝐵= triangular numbers
𝐶= 1 less than multiple of 4
Construct a Venn Diagram for these sets. 3 ? 𝑨 4 2 5 ? 7 10 3 8 ? 1 6 𝑩 𝑪

12. #2 :: Venn Diagrams involving Frequencies
Sometimes we just have one number in each region, representing the number of items in that region, rather than the items themselves.
A vet surveys 100 of her clients. She finds that
25 own dogs, 15 own dogs and cats, 11 own dogs and tropical fish, 53 own cats, 10 own cats and tropical fish, 7 own dogs, cats and tropical fish, 40 own tropical fish.
Fill in this Venn Diagram, and hence answer the following questions:
𝑃 𝑜𝑤𝑛𝑠 𝑑𝑜𝑔 𝑜𝑛𝑙𝑦
𝑃 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑜𝑤𝑛 𝑡𝑟𝑜𝑝𝑖𝑐𝑎𝑙 𝑓𝑖𝑠ℎ
𝑃(𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑜𝑤𝑛 𝑑𝑜𝑔𝑠, 𝑐𝑎𝑡𝑠, 𝑜𝑟 𝑡𝑟𝑜𝑝𝑖𝑐𝑎𝑙 𝑓𝑖𝑠ℎ)
Fro Tip: The trick is to start from the centre of the diagram and work outwards. 𝝃 𝑪 ? 35 ? 11
6 100
60 100
11 100 ? ? ? 8 3 ? 𝑫 ? 7 ? 𝑭 ? ? ? 26 6 4
Dr Frost’s cat “Pippin”

13. #2 :: Venn Diagrams involving Frequencies
Conditional Probabilities
Given that a randomly chosen person owns a cat, what’s the probability they own a dog? ?
Now our choice is being restricted just to those who owns cats (i.e. 53).
Thus the probability is 𝝃 𝑪 35 11 8 3 𝑫 7 𝑭 26 6 4
Dr Frost’s cat “Pippin”

14. Test Your Understanding
Edexcel S1 Jan 2012 Q6
The following shows the results of a survey on the types of exercise taken by a group of 100 people.
65 run 48 swim
60 cycle 40 run and swim
30 swim and cycle 35 run and cycle 25 do all three
(a) Draw a Venn Diagram to represent these data (4)
Find the probability that a randomly selected person from the survey
(b) takes none of these types of exercise, (2)
(c) swims but does not run, (2)
takes at least two of these types of exercise (2)
Jason is one of the above group. Given that Jason runs,
(e) find the probability that he swims but does not cycle (3) ? ? ? ? ?
3 13

15. 𝜉 Exercise 2 ? Orc B 32 students 48 12 8 32 Question 1
(on provided sheet)
Question 1
[JMC 2002 Q11] The Pythagoras School of Music has 100 students. Of these, 60 are in the band and 20 are in the orchestra. Given that 12 students are in both the band and the orchestra, how many are in neither the band nor the orchestra? ? 𝜉
Orc B
32 students 48 12 8 32

16. Exercise 2
(on provided sheet) 2 ? ? ? ?

17. Exercise 2
(on provided sheet) 3 ? ? ? ?

18. Exercise 2
(on provided sheet) 4 ? ? ? ?

19. Exercise 2
(on provided sheet) 5 ? ? ? ? ? ?

20. Exercise 2
(on provided sheet)
a ? 6
b ?
c ?
d ?
e ?

21. Exercise 2
(on provided sheet) 7
a ?
b ?
c ?
d ?
e ?

22. Exercise 2 ? (on provided sheet) N
[SMC 2011 Q17] Jamie conducted a survey on the food preferences of pupils at a school and discovered that 70% of the pupils like pears, 75% like oranges, 80% like bananas and 85% like apples.
What is the smallest possible percentage of pupils who like all four of these fruits?
At least 10%
At least 15%
At least 20%
At least 25%
At least 70% ?
For now just think of the ∩ symbol as meaning “and”. We’ll see it next lesson.

23. #3 Combining Sets 𝐴∩𝐵={3,4} ? 𝐴∪𝐵={1,2,3,4,5,6,7} ? 9 2 5 1 3 6 4 7 8
We have various operations on numbers, such as addition: 1+2=3 and multiplication: 2×3=6
So are there similar operations on sets? Yes! 𝝃 9 𝑩 𝑨 2 5 1 3 6 4 7 8 10
𝐴∩𝐵={3,4} ?
𝐴∪𝐵={1,2,3,4,5,6,7} ?
! 𝐴∩𝐵 is the intersection of 𝐴 and 𝐵
It means “the things in A and in B”
! 𝐴∪𝐵 is the union of 𝐴 and 𝐵
It means “the things in A or in B”*
* Things in A or B also includes things in both. Things in one but not the other is known as “exclusive or”. You do not need to know this!

24. #3 Combining Sets 𝐴′={5,6,7,8,9,10} ? 9 2 5 1 3 6 4 7 8 10 𝝃 𝑩 𝑨
We have various operations on numbers, such as addition: 1+2=3 and multiplication: 2×3=6
So are there similar operations on sets? Yes! 𝝃 9 𝑩 𝑨 2 5 1 3 6 4 7 8 10
𝐴′={5,6,7,8,9,10} ?
𝐴′ is the complement of 𝐴
It means “the things not in A”

25. Quickfire Examples 𝜉= 1,2,3,…,10 𝐴= 2,4,6,8,10 𝐵={3,6,9}𝝃 𝑨 𝑩
𝜉= 1,2,3,…,10 𝐴= 2,4,6,8,10 𝐵={3,6,9} 𝑎 𝑏 𝑐 𝑒 𝑑
𝐴∪𝐵= 2,3,4,6,8,9,10 𝐴∩𝐵= 6 𝐴 ′ = 1,3,5,7,9 𝐴∩ 𝐵 ′ = 2,4,8,10 𝐴 ′ ∩𝐵={3,9} 𝐴 ′ ∩ 𝐵 ′ = 1,5,7 ?
𝐴∩𝐵= 𝑏,𝑑 𝐴∪𝐵= 𝑎,𝑏,𝑐,𝑑,𝑒 𝐴 ′ = 𝑐 𝐵 ′ = 𝑎,𝑒 𝐴∩ 𝐵 ′ = 𝑎,𝑒 𝐴 ′ ∩𝐵={𝑐} 𝐴 ′ ∩ 𝐵 ′ =∅ ? ? ? ? ? ?
read this as “A and not B” ? ? ? ? ? ?
There is nothing that is “not in A and not in B”!
∅ is the empty set, as seen earlier.

26. Test Your Understanding
(on provided sheet) 1 2
𝜉={ all whole numbers }
𝐴={ factors of 60 } 𝐵={ multiples of 3 }
List the members of the set 𝐴∩𝐵. ?
𝐴∩𝐵={3,6,12,15,30,60} a ?
10,12,14,15,16,18 b 7 10 ?

27. Matching Game! (on provided sheet)
Match the set expressions with the indicated regions.
Click an expression below to reveal. 𝑨 𝝃 𝑩 𝑪 𝑨 𝝃 𝑩 𝑪 𝑨 𝝃 𝑩 𝑪
𝐴∩𝐵′
𝐴∩ 𝐵 ′ ∩𝐶′ 𝑨 𝝃 𝑩 𝑪 𝑨 𝝃 𝑩 𝑪 𝑨 𝝃 𝑩 𝑪
𝐴∪𝐵
𝐴 ′ ∩𝐵′
𝐴∩𝐵 𝑨 𝝃 𝑩 𝑪 𝑨 𝝃 𝑩 𝑪 𝑨 𝝃 𝑩 𝑪
𝐴∩(𝐵∩𝐶)′
𝐴∪𝐵 ∩ 𝐴∩𝐵∩𝐶 ′
𝐴∩𝐵∩𝐶′
𝐴∪𝐵 ∩𝐶′

28. Cardinality of Sets ? 𝑛 𝐴 =4 𝑛 𝐵 =5 𝑛 𝐴∩𝐵 =2 𝑛 𝐴 ′ ∩𝐵 =3 ? ? ?
! The cardinality of a set is the size of the set. Use 𝑛(𝐴) or |𝐴| for the cardinality/size of a set 𝐴. 𝝃 9 𝑩 𝑨 2 5 1 3 6 4 7 8 10
𝑛 𝐴 =4 𝑛 𝐵 =5 𝑛 𝐴∩𝐵 =2 𝑛 𝐴 ′ ∩𝐵 =3 ? ? ? ?

29. Exercise 3
(on provided sheet) 1 𝝃 2 𝝃 𝑨 𝑩 𝑨 𝑩 𝑞 𝑡 4 𝑝 8 13 𝑟 𝑢 1 𝑠 11 𝑥 6 𝑤 𝑣
List the numbers in:
𝐴∩𝐵={𝟏,𝟖}
𝐴∪𝐵= 𝟏,𝟒,𝟖,𝟏𝟏,𝟏𝟑
𝐴 ′ ={𝟔,𝟏𝟏,𝟏𝟑}
𝐴 ′ ∩𝐵={𝟏𝟏,𝟏𝟑}
Given that a number is chosen at random, find the probability it is in set 𝐵.
𝟐 𝟑 𝑦 𝑪 a ? b ?
Determine the sets:
𝐴∩𝐵={𝒒,𝒓,𝒔}
𝐴∪𝐶= 𝒑,𝒒,𝒓,𝒔,𝒗,𝒙
𝐴 ′ ∩𝐶′={𝒕,𝒖,𝒘,𝒚}
𝐴∩𝐵∩𝐶={𝒔}
𝐴∩ 𝐵 ′ ∩𝐶=∅
𝐴∪𝐶 ∩ 𝐵 ′ ={𝒑,𝒗}
Given that a number is chosen at random from the set 𝐴∪𝐶, find the probability it is in set 𝐵.
4 6 c ? a ? d ? b ? c ? e d ? e ? f ? ? g ?

30. Exercise 3 ? ? ? ? ? ? ? ? ? 11 3 5 1 𝝃 𝑩 𝑨 (on provided sheet) 3
𝐴 and 𝐵 are two sets such that:
𝑛 𝜉 =20 𝑛 𝐴 =14 𝑛 𝐴∩𝐵 =3 𝑛 𝐵 ′ =12
Form a Venn diagram, where the number in each region is the number of elements in it. 4
𝐴= 𝑓,𝑟,𝑜,𝑠,𝑡 𝐵={𝑏,𝑎,𝑟,𝑡,𝑜,𝑛}
Determine the sets:
𝐴∩𝐵= 𝒐,𝒓,𝒕 𝐴∪𝐵= 𝒇,𝒓,𝒐,𝒔,𝒕,𝒃,𝒂,𝒏 𝐴 ′ ∩𝐵={𝒃,𝒂,𝒏} a ? b ? c ?
𝜉= 1,2,3,4,5,6,7,8,9,10 𝐴={ all prime numbers }
𝐵={ all multiples of 6 }
Determine:
𝐴∪𝐵= 𝟐,𝟑,𝟓,𝟔,𝟕 𝐴∩𝐵=∅ 5 𝝃 ? 𝑨 𝑩 ? a 11 3 5 b ?
𝜉= 1,2,3,4,5,6,7,8,9,10 𝐴={ all even numbers }
𝐵={ all factors of 8 }
Determine:
𝐴∪𝐵= 𝟏,𝟐,𝟒,𝟔,𝟖 𝐴∩𝐵= 𝟐,𝟒,𝟖 𝐴∩ 𝐵 ′ = 𝟔,𝟏𝟎 1 6 a ? b ? c ?

31. Exercise 3 ? ? ? 𝝃 𝑪 𝑨 𝑩 𝝃 𝑪 𝑨 𝑩 (on provided sheet)
Construct a Venn Diagram for sets 𝐴,𝐵,𝐶 (without numbers) such that:
𝐴⊂𝐵, 𝐵∩𝐶=∅ 7 N
𝐴,𝐵,𝐶 are sets such that:
𝑛 𝐴 =20, 𝑛 𝐵 =20, 𝑛 𝐶 =20, 𝑛 𝐴∩𝐵 =10 𝑛 𝐴∩𝐶 =10 𝑛 𝐵∩𝐶 =10 𝑛 𝐴∩𝐵∩𝐶 =5
Determine 𝑛(𝐴∪𝐵∪𝐶).
By adding 𝒏 𝑨 ,𝒏 𝑩 ,𝒏(𝑪), we’re double counting regions which overlap once, and triple counting where all three sets overlap. Subtracting 𝒏 𝑨∩𝑩 ,𝒏 𝑨∩𝑪 ,𝒏(𝑩∩𝑪), now all regions are counted once, but we’re missing 𝑨∩𝑩∩𝑪, so add it back on.
𝟐𝟎+𝟐𝟎+𝟐𝟎−𝟏𝟎−𝟏𝟎−𝟏𝟎+𝟓 =𝟑𝟓
This is known as the inclusion-exclusion principle. a 𝝃 ? 𝑪 𝑨 𝑩 b
𝐴⊂𝐵, 𝐴∩𝐶=∅ ? 𝝃 ? 𝑪 𝑨 𝑩