Set Notation

Venn Diagrams or “Circle Diagram” displays information in a simple way to show overlaps

between the sets of data.
A set is a collection of things.

An element is one item in a set.
e.g. A = {a, e, I, o, u}
curly brackets
capital letter








elements
Set A has 5 elements n(A) = 5
or
ǀAǀ = 5
a is an element of A
a ϵ A
b is not an element of A
b ϵ A
{ a, e } is a subset of A
{a, e} ϲ A
{b, e} is not a subset of A
{b, e} ϲ A
We usually list the elements inside the curly brackets in alphabetical or numerical order (but
is not essential)
If a set has no elements, i.e. it is empty, we write Ø for the empty set.
e.g.
{Multiples of 2 which are odd} = Ø
All the elements under consideration belong to a set called the universal set, U
e.g.
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
U
A
2
3
5
7
4



A = {2, 3, 5, 7}
B
6
8
B = {1, 3, 5, 7, 9}
1 9
10
The overlap between two sets is called the intersection between two sets (∩)
e.g. above:
A ∩ B = {3, 5, 7}
A
B
This is often written as the elements
in set A AND in set B
The combination of elements of one set and another set is called the union (∪)
e.g. above:
A ∪ B = {1, 2, 3, 5, 7, 9}
A
B
This is often written as the elements
in set A OR in set B
The elements lying in the Universal set but not in a particular set are called the
complement
e.g. above:
A
since A = {2, 3, 5, 7}
A’ or 𝐴̅ = {1, 4, 6, 8, 9, 10}
A
B
or
Example 1
List:
U
B
A
a
b
c
k
e
h
l
k
i
j
m
d
g
a) U
b) C
c) D
State: j) n(C)
̅)
k) n(𝐷
d) C’
̅
e) 𝐷
f) 𝐶 ∩ 𝐷
l) n(𝐶 ∪ 𝐷)
m) n(𝐶 ∩ 𝐷)
g) 𝐶 ∪ 𝐷
h) 𝐶 ∩ 𝐷′
i) ̅̅̅̅̅̅̅̅
𝐶∪𝐷
̅̅̅̅̅̅̅̅
n) n(𝐶
∩ 𝐷)
o) n(𝐶̅ ∩ 𝐷)
Solution
a) U = {a, b, c, d, e, f, g, h, I, j, k, l, m}
b) C = {a, b, f, h, k}
c) D = {b, c, e, f, l}
d) C’ = {c, d, e, g, I, j, l, m}
̅ = {a, d, g, h, I, j, k, m}
e) 𝐷
j) 5
k) 8
𝐶 ∩ 𝐷 = {b, f}
𝐶 ∪ 𝐷 = {a, b, c, e, f, h, k, l}
𝐶 ∩ 𝐷′ = {a, h, k}
̅̅̅̅̅̅̅̅
𝐶 ∪ 𝐷 = {d, g, i, j, m}
m) 2 n) 11 o) 3
f)
g)
h)
i)
l) 8
U
Example 2
A
B
Draw copies of this Venn Diagram and shade the region indicated.
b) 𝐴 ∩ 𝐵
e) (𝐴 ∩ 𝐵) ∩ 𝐶)
h) 𝐴 ∩ (𝐵 ∪ 𝐶)
a) A
d) B’
g) 𝐴 ∪ (𝐵 ∩ 𝐶)
c)
f)
i)
𝐴∪𝐶
𝐴∪𝐵∪𝐶
𝐴̅ ∩ (𝐵 ∪ 𝐶)
C
Solution
a)
b)
U
c)
U
A
U
A
B
C
A
B
C
d)
C
f)
e)
U
U
A
B
U
A
C
A
B
C
g)
i)
U
A
B
B
C
h)
U
B
U
A
C
B
A
C
C
Using Venn Diagrams to solve problems
Example 1
n(𝐴 ∩ 𝐵) = 5
n(A) = 12
n(B) = 18
Display on a Venn Diagram and fill in the empty spots.
B
n(U) = 31
𝑛(𝐴) − 𝑛(𝐴 ∩ 𝐵) = 12 − 5 = 7
U=31
A
12
B
18
Solution
7
5
13
𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵) = 18 − 5 = 13
6
Sadler 3A Exercise 3A
𝑛(𝑈) − 𝑛(𝐴 ∪ 𝐵) = 31 − (7 + 5 + 13) = 6
Combining Probability and Sets
Example 1
̅̅̅̅̅̅̅
n(U) = 60
n(A) = 20
n(B) = 25
n(𝐴
∪ 𝐵 = 30
Find the probability that, if one person was selected at random form the universal set, U, they are in:
a) set A
b) 𝐴 ∪ 𝐵
c) B’
d) set A given that they are in set B
Solution
U
20
A
B
5
b) P (selecting a person in 𝐴 ∪ 𝐵) =
10
15
𝑛(𝐴)
𝑛(𝐵)
30
𝑛(𝐴∪𝐵)
= 𝑛(𝑈)
60
a) P (selecting a person in set A) = 60 =
30
35
60
c)
P (B’) =
d)
P (A ǀ B ) =
15
25
=
𝑛(𝐴∩𝐵
𝑛(𝐵)
Four ways of using Venn Diagrams – the following all show the same example
This shows the number of elements in each
section
E.g.
19 + 15 = n(A)
15 + 12 = n(B)
19 + 15 + 12 = n(𝐴 ∪ 𝐵)
19 + 15 + 12 + 4 = 50 = n(U)
U=50
B
A
19
15
12
4
Using fractions to denote probabilities:
19 15 12 4
50
+
+
+
=
=1
50 50 50 50 50
U=1
B
A
19
50
15
50
12
50
E.g. P(A) =
4
50
19+15
50
15
P(𝐴 ∩ 𝐵) = 50
34
= 50
P(B) =
15
15+12
50
P(A ǀ B) = 15+12 =
=
27
50
15
27
U=1
B
A
0.38
0.3
Using decimals to denote probabilities:
0.38 + 0.3 + 0.24 + 0.08 = 1
E.g.
P(A’) = 0.24 + 0.08 = 0.32
P(𝐴 ∪ 𝐵) = 0.38 + 0.3 + 0.24 = 0.92
0.24
0.08
U=
100%
B
A
38%
30%
24%
8%
Using percentages to denote probabilities:
38% + 30% + 24% + 8% = 100%
̅̅̅̅̅̅̅
E.g.
P(𝐴
∩ 𝐵) = (38+24+8)% = 70%
̅̅̅̅̅̅̅
P(𝐴
∪ 𝐵) = 8%
Take care to read the question properly to see which of the above you should use.
Do Sadler 3A Exercise 3B
Sample Spaces
The list of all possible outcomes that can occur when an experiment or event is carried out is called a
sample space. Three ways of showing a sample space are outlined below.
Note: We are referring here to equally likely outcomes. Each event has the same chance of
occurring.
1. List
Used when you have a small sample space.
E.g. Rolling a die once {1, 2, 3, 4, 5, 6}
1
P(2) = 6
2. Table
E.g. Rolling two dice and looking at the total.
1
2
3
4
5
6
1
2
3
4
5
6
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
P(total of 5) =
4
36
10
P(more than 8) = 36
3. Tree Diagrams
A meal
- Starter of prawn cocktail or soup
- Mains of fish, beef, chicken or vegetarian
- Desert of gateaux, ice-cream or mudcake
Restrictions
- Can’t have fish if you had prawn cocktail
- Beef must be followed by ice-cream
The above information can be transferred onto a tree diagram as follows:
STARTER
MAIN
B
DESERT
I
C
P
V
F
B
S
C
V
SAMPLE SPACE
PBI
G
PCG
I
PCI
M
PCM
G
PVG
I
PVI
M
PVM
G
SFG
I
SFI
M
SFM
I
SBI
G
SCG
I
SCI
M
SCM
G
SVG
I
SVI
M
SVM
17 in the
sample space
E.g.
P(I) =
7
17
P(I ǀ P) =
Two Way Tables
Example
The table shows the results of 1000 votes in the Mandurah community to poll opinions about
whether to finish school 30 minutes later.
a) Complete the table above using the
following information. The teachers
Yes
No
Undecided Total
who responded ‘yes’ represent two
Student
24
21
10
thirds of the students who responded
Teacher
3
‘yes’, while the undecided parents
Parent
11
6
3
represented only half of the total of
Total
100
undecided teachers.
b) For a randomly surveyed person,
determine the probability that the person:
i)
was a teacher in favour of finishing later.
ii)
was a student.
3
7
iii)
iv)
v)
vi)
vii)
Solution
Student
Teacher
Parent
Total
was not in favour of finishing later.
was a parent given that the person was in favour of finishing later.
was not in favour of finishing later if the person was a student.
was not a parent if the person did not want to finish later.
was either not in favour or undecided in finishing later given the person was a
teacher.
a)
Yes
24
16
11
51
No
21
3
6
30
Undecided Total
10
55
6
25
3
20
19
100
REVIEW
b) i)
16
100
= 25
4
iii)
30
100
= 10
v)
21
55
3
vi)
ii)
55
100
iv)
11
51
24
30
4
=5
11
= 20
v)
9
25