2.2  Set Notation and Operations
2.2  Set Notation and OperationsOutcome Space
When you first start learning about sets and events, it is often helpful to consider the outcome space.
 Outcome Space
 The outcome space of a scenario is all the possible outcomes that can occur and is often denoted S. The outcome space may also be referred to as the sample space
Example 22
Consider the experiment where two fair sixsided dice are rolled and their face values recorded. Write down the outcome space.
Answer
If we write the pair of faces such as (value of first die, value of second die), then...
First Die  

1  2  3  4  5  6  
Second Die 
1  1, 1  2, 1  3, 1  4, 1  5, 1  6, 1 
2  1, 2  2, 2  3, 2  4, 2  5, 2  6, 2  
3  1, 3  2, 3  3, 3  4, 3  5, 3  6, 3  
4  1, 4  2, 4  3, 4  4, 4  5, 4  6, 4  
5  1, 5  2, 5  3, 5  4, 5  5, 5  6, 5  
6  1, 6  2, 6  3, 6  4, 6  5, 6  6, 6 
In set notation we can write...
S = {(1,1) (2,1) (3,1) (4,1) (5,1) (6,1) (1,2) (2,2) (3,2) (4,2) (5,2) (6,2) (1,3) (2,3) (3,3) (4,3) (5,3) (6,3) (1,4) (2,4) (3,4) (4,4) (5,4) (6,4) (1,5) (2,5) (3,5) (4,5) (5,5) (6,5) (1,6) (2,6) (3,6) (4,6) (5,6) (6,6)}
There are 36 possible outcomes in the sample space S.
Try It! Outcome Spaces

Getting an even number on the face of the second die.
Let A = {an even number on the face of the second die}.
First Die 1 2 3 4 5 6 Second
Die
1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 3 1, 3 2, 3 3, 3 4, 3 5, 3 6, 3 4 1, 4 2, 4 3, 4 4, 4 5, 4 6, 4 5 1, 5 2, 5 3, 5 4, 5 5, 5 6, 5 6 1, 6 2, 6 3, 6 4, 6 5, 6 6, 6 In set notation...
A = {(1, 2), (1, 4), (1, 6), (2,2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6), (4, 2), (4, 4), (4, 6), (5, 2), (5, 4), (5, 6), (6, 2), (6, 4), (6, 6)}

The sum of two faces is greater than or equal to 10.
Let B = {sum of the two faces is greater than or equal to 10}
First Die 1 2 3 4 5 6 Second
Die
1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 3 1, 3 2, 3 3, 3 4, 3 5, 3 6, 3 4 1, 4 2, 4 3, 4 4, 4 5, 4 6, 4 5 1, 5 2, 5 3, 5 4, 5 5, 5 6, 5 6 1, 6 2, 6 3, 6 4, 6 5, 6 6, 6 In set notation...
B = {(6, 4), (5, 5), (6, 5), (4, 6), (5, 6), (6,6)}
Set Operations
Now that we know how to denote events, the next step is to use the set notation to represent set operations. Each operation will also be presented in a Venn Diagram. Set operations are important because they allow us to create a new event by manipulation of other events
 Union

Verbally: The union of two events, A and B, contains all of the outcomes that are in A, B or both. In statistics, ‘or’ means at least one event occurs and therefore includes the event where both occur,
Symbolically: The union of A and B is denoted $A\cup B$.
Visually: A or B also written as \(A \cup B\) = {outcomes in both A or B or both}
 Intersection

Verbally: The intersection of two events, A and B, contains all of the outcomes that are in both A and B.
Symbolically: The intersection is denoted by $A \cap B$
Visually: A and B also written as \(A \cap B\) = {outcomes in both A and B}
 Complement

Verbally: The complement of an event, A, contains all of the outcomes that are not in A.
Symbolically: The complement can be denoted as $A^c$, $\bar{A}$, or $A^\prime$.
Visually: A^{'} also written as \( \bar{A}\) or $A^c$ = {outcomes not in A}
 Mutually Exclusive

Verbally: A and B are called mutually exclusive (or disjoint) if the occurrence of outcomes in A excludes the occurrence of outcomes in B. One example of two mutually exclusive events is A and A'.
Symbolically: There are no elements in \(A \cap B\) and thus \(A \cap B=\emptyset \) . The empty set, denoted as $\emptyset$, is an event that contains no outcomes.
Visually: \(A \cap B=\emptyset \)
Example 23
Let's go back to the example where we roll two fair sixsided die. Given the following events:
 \(A={(3, 5)}\)
 \(B=\text {a 4 is rolled on the first die}\)
 \(C=\text {a 5 is rolled on the second die}\)
 \(D=\text {the sum of the dice is 7}\)
 \(E={(7, 4)}\)
Find $B\cap D$ and $B\cup D$
Answer
$B\cap D=\{(4,3)\}$
$B\cup D=\{(4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6, 1), (5, 2), (3, 4), (2, 5), (1, 6)\}$
Visually,
First Die  

1  2  3  4  5  6  
Second Die 
1  1, 1  2, 1  3, 1  4, 1  5, 1  6, 1 
2  1, 2  2, 2  3, 2  4, 2  5, 2  6, 2  
3  1, 3  2, 3  3, 3  4, 3  5, 3  6, 3  
4  1, 4  2, 4  3, 4  4, 4  5, 4  6, 4  
5  1, 5  2, 5  3, 5  4, 5  5, 5  6, 5  
6  1, 6  2, 6  3, 6  4, 6  5, 6  6, 6 
Try It! Outcome of Sets

$D\cap C$ and $D\cup C$$D\cap C=\{(2, 5)\}$, $D\cup C=\{(1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (6,1), (5, 2), (4, 3), (3, 4), (1, 6)\}$

$A\cap D$ and $A\cup D$$A\cap D=\{\emptyset\}$, $A\cup D=\{(6, 1), (5,2), (4,3), (3, 4), (2, 5), (1, 6), (3, 5)\}$

$B\cap C$ and $B\cup C$$B\cap C={(4, 5)}$, $B\cup C=\{(4, 1), (4, 2), (4,3), (4,4), (4, 5), (4, 6), (1,5), (2, 5), (3, 5), (5, 5), (6,5)\}$
Example 24
Suppose events $A$, $B$, and $C$ are events of a particular scenario. Write the following using event notation.
 At least one event occurs.
Answer: $A\cup B\cup C$ At least one event means A or B or C or any two events or all three events.
 None of the events occur.
Answer: $A^\prime\cap B^\prime \cap C^\prime$
 Only A occurs.
Answer: $A\cap (B\cup C)^\prime$