GCSE Sets
Dr J Frost ([email protected])
www.drfrostmaths.com
@DrFrostMaths
Last modified: 3rd July 2017
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)
What is a set?
Is it a set?
−3.5, 2, 9
4, 5, 5, 6
1
A set with one thing in it is
known as a singleton.
It’s possible to
have a set of sets!
1,2 , 3,4
𝑟𝑒𝑑, 𝑏𝑙𝑢𝑒, 𝑔𝑟𝑒𝑒𝑛
False
True
False
True
False
True
False
True
False
True
False
True
Sets need not consist of numbers.
Are these sets the same?
3,1,2 = {1,2,3}
Finite Sets vs Infinite Sets
−4, 1, 3
The examples with seen have
been finitely large sets.
But it’s also possible to have sets which
are infinitely large…
“the set of all positive integers
(whole numbers)”
“the set of all odd numbers”
(At A Level and beyond,
the symbol ℕ is used for
such a set)
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”
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.
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.
𝜉
What does this region represent?
The items in 𝑨?
and in 𝑩.
𝐴
𝐵
6
2
1
8
10
0
3
−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”)
?
Example
𝝃 = whole numbers
from 1 to 15
𝜉
𝐴
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
7
8
9
15
𝐵
𝐶
Test Your Understanding
𝝃 = whole numbers
from 1 to 10
𝜉
𝐴
𝑨 = set of all cube
numbers
8
2
𝑩 = set of all odd
numbers
4
𝑪 = set of all multiples
of 3
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
?
10
1
? Venn Diagram
9
5
7
𝐵
3
6
𝐶
Further Examples
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
𝜉 = 1,2,3,4,5,6
𝐴 = 1,2,3
𝐵= 1
Construct a Venn Diagram to
show these sets.
𝝃
𝑨
𝑩
3
2
1
4?
𝑨
3
2
5
6
𝑩
1?
4
5
6
Any number in 𝐵 is also in 𝐴. It would therefore be a
good idea to draw 𝐵 inside 𝐴 to show this relationship.
Elements of Sets and Subsets
An ‘element’ or ‘member’ is an item in a set.
!
2 ∈ 1,2
3 ∉ 1,2
𝐴⊂𝐵
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”.
means that 3 is not a member of
the set.
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
False
True
False
True
2,3,4 ⊂ 3,4,5
False
True
0,2,3 ⊂ {−1,0,1,2,3,4}
False
True
2∉𝑃
(where 𝑃 is the set of
all prime numbers)
Exercise 1
1
𝜉 = 4,5,6,7,8 ,
𝐵 = 6,7,8
𝐴 = 5,6,7
4
Construct a Venn Diagram to show these sets.
𝝃 𝑨
2
5 67? 8
𝝃
𝑩4
5
Construct a Venn Diagram to show these sets.
𝑨
2
3 8?
𝑩
4
3 𝜉 = 1,2, … , 10
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
𝑩
𝑪
𝑨
7
1
6
𝐴 = set of all primes
𝐵 = triangular numbers
𝐶 = 1 less than multiple of 4
Construct a Venn Diagram for
these sets.
?𝑩
𝑨
𝜉 = 1,2,3,4,5,6,7,8
𝐴 = 1,2,3,8 , 𝐵 = {3,4,5,8}
𝐶 = 1,5,6,8
𝝃
You have two sets 𝐴 and 𝐵 and 𝐵 ⊂ 𝐴. Draw a Venn
Diagram (without any numbers) that indicates the
relationship between the sets.
𝑪
N
4
𝑨 2
5
3 7
10 ?
8
16
𝑪
𝑩
?
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
a) Determine 𝑃 1,2,3
= ∅, 𝟏 , 𝟐 , 𝟑 , 𝟏, 𝟐 , 𝟏, 𝟑 , 𝟐, 𝟑 , 𝟏, 𝟐, 𝟑
b) 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 𝟐𝒏 possible subsets.
?
?
#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.
Fro Tip: The trick is to start
from the centre of the
diagram and work outwards.
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:
a) 𝑃 𝑜𝑤𝑛𝑠 𝑑𝑜𝑔 𝑜𝑛𝑙𝑦
b) 𝑃 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑜𝑤𝑛 𝑡𝑟𝑜𝑝𝑖𝑐𝑎𝑙 𝑓𝑖𝑠ℎ
c) 𝑃(𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑜𝑤𝑛 𝑑𝑜𝑔𝑠, 𝑐𝑎𝑡𝑠, 𝑜𝑟 𝑡𝑟𝑜𝑝𝑖𝑐𝑎𝑙 𝑓𝑖𝑠ℎ)
𝝃
𝑪
?
35
a)
b)
c)
6
100
60
100
11
100
?
?
?
?
8
𝑫
?6
Dr Frost’s cat
“Pippin”
?7
?4
11
?
?3
?
26
𝑭
#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).
?
15
Thus the probability is 53.
𝝃
𝑪
35
8
𝑫
3
7
6
Dr Frost’s cat
“Pippin”
11
4
26
𝑭
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)
(d) 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
?
Exercise 2
(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
48
12
32
8 ?
32
students
Exercise 2
(on provided sheet)
2
?
?
?
?
Exercise 2
(on provided sheet)
3
?
?
?
?
Exercise 2
(on provided sheet)
4
?
?
?
?
Exercise 2
(on provided sheet)
5
?
?
?
?
?
?
Exercise 2
(on provided sheet)
6
a?
b?
c?
d?
e?
Exercise 2
(on provided sheet)
7
a?
b?
c?
d?
e?
Exercise 2
N
(on provided sheet)
[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?
A.
B.
C.
D.
E.
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.
#3 Combining Sets
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
𝑨
𝑩
2
1
8
𝐴 ∩ 𝐵 = {3,4}
?
! 𝐴 ∩ 𝐵 is the intersection of 𝐴 and 𝐵
It means “the things in A and in B”
3
4
5
6
7
10
𝐴 ∪ 𝐵 = {1,2,3,4,5,6,7}
?
! 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!
#3 Combining Sets
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
𝑨
𝑩
2
1
3
4
8
5
6
7
𝐴′ = {5,6,7,8,9,10}
?
! 𝐴′ is the complement of 𝐴
It means “the things not in A”
10
Quickfire Examples
𝝃 𝑨
𝑩
𝑎
𝑒
𝑏
𝑑
𝑐
𝐴 ∩ 𝐵 = 𝑏,?𝑑
𝐴 ∪ 𝐵 = 𝑎, 𝑏, ?𝑐, 𝑑, 𝑒
𝐴′ = 𝑐 ?
read this as “A
and not B”
𝐵′ = 𝑎, 𝑒?
𝐴 ∩ 𝐵′ = 𝑎, 𝑒?
𝐴′ ∩ 𝐵 = {𝑐} ?
is nothing that is
𝐴′ ∩ 𝐵′ = ∅? There
“not in A and not in B”!
∅ is the empty set, as
seen earlier.
𝜉 = 1,2,3, … , 10
𝐴 = 2,4,6,8,10
𝐵 = {3,6,9}
?
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 ?
Test Your Understanding
1
(on provided sheet)
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
Matching Game!
(on provided sheet)
Match the set expressions with the indicated regions.
𝝃
𝝃
𝑨
𝝃
𝑨
Click an expression
below to reveal.
𝑨
𝐴 ∩ 𝐵′
𝑩
𝝃
𝑪
𝑩
𝝃
𝑨
𝑪
𝑩
𝝃
𝑨
𝐴 ∩ 𝐵′ ∩ 𝐶′
𝑪
𝑨
𝐴∪𝐵
𝑩
𝝃
𝑪
𝑩
𝝃
𝑨
𝑪
𝑩
𝝃
𝑨
𝑪
𝐴′ ∩ 𝐵′
𝐴∩𝐵
𝐴 ∩ (𝐵 ∩ 𝐶)′
𝑨
𝐴∪𝐵 ∩ 𝐴∩𝐵∩𝐶
𝑩
𝑪
𝑩
𝑪
𝑩
𝑪
𝐴 ∩ 𝐵 ∩ 𝐶′
𝐴 ∪ 𝐵 ∩ 𝐶′
′
Cardinality of Sets
! 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
𝑨
𝑩
2
1
8
𝑛
𝑛
𝑛
𝑛
3
4
5
6
7
𝐴 =4?
𝐵 =5?
𝐴∩𝐵 =2?
𝐴′ ∩ 𝐵 = 3 ?
10
Exercise 3
𝝃
1
𝑨
6
a
b
c
d
4
(on provided sheet)
𝑩
𝝃 𝑨
2
8 13
1 11
List the numbers in:
𝐴 ∩ 𝐵 = {𝟏, 𝟖} ?
𝐴 ∪ 𝐵 = 𝟏, 𝟒, 𝟔, 𝟖,
? 𝟏𝟏, 𝟏𝟑
′
𝐴 = {𝟔, 𝟏𝟏, 𝟏𝟑}
?
′
𝐴 ∩ 𝐵 = {𝟏𝟏, 𝟏𝟑}?
e Given that a number is
chosen at random, find the
probability it is in set 𝐵.
𝟐
𝟑?
𝑤
𝑝
𝑣
𝑦
a
b
c
d
e
f
𝑩
𝑞 𝑡
𝑟 𝑢
𝑠
𝑥
𝑪
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} ?
g Given that a number is chosen at
random from the set 𝐴 ∪ 𝐶, find the
probability it is in set 𝐵.
1
?2
Exercise 3
3
(on provided sheet)
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.
𝝃
5
11
?3
𝐴 = 𝑓, 𝑟, 𝑜, 𝑠, 𝑡
𝐵 = {𝑏, 𝑎, 𝑟, 𝑡, 𝑜, 𝑛}
Determine the sets:
a 𝐴 ∩ 𝐵 = 𝒐, 𝒓, 𝒕 ?
b 𝐴 ∪ 𝐵 = 𝒇, 𝒓, 𝒐, 𝒔,?𝒕, 𝒃, 𝒂, 𝒏
c 𝐴′ ∩ 𝐵 = {𝒃, 𝒂, 𝒏} ?
𝜉 = 1,2,3,4,5,6,7,8,9,10
𝐴 = { all prime numbers }
𝐵 = { all multiples of 6 }
Determine:
a 𝐴 ∪ 𝐵 = 𝟐, 𝟑, 𝟓, 𝟔,?𝟕
b 𝐴∩𝐵 =∅
?
𝑩
𝑨
1
4
5
6
𝜉 = 1,2,3,4,5,6,7,8,9,10
𝐴 = { all even numbers }
𝐵 = { all factors of 8 }
Determine:
a 𝐴 ∪ 𝐵 = 𝟏, 𝟐, 𝟒,
𝟔, 𝟖
?
c 𝐴 ∩ 𝐵′ = 𝟔, 𝟏𝟎?
b 𝐴 ∩ 𝐵 = 𝟐, 𝟒,?𝟖
Exercise 3
7
(on provided sheet)
Construct a Venn Diagram for
sets 𝐴, 𝐵, 𝐶 (without numbers)
such that:
a
𝐴 ⊂ 𝐵,
𝐵∩𝐶 =∅
𝝃
𝑨
𝑩
b
𝑪
?
𝐴 ⊂ 𝐵,
𝝃
𝑨
𝑩
𝐴∩𝐶 =∅
𝑪
?
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.