2.2 - Set Notation and Operations

2.2 - Set Notation and Operations

Outcome 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 2-2

Consider the experiment where two fair six-sided dice are rolled and their face values recorded. Write down the outcome space.

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

Directions: Write out the event in probability notation and then identify the outcome space.
1. 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)}

2. 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 2-3

Let's go back to the example where we roll two fair six-sided 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 $B\cup D$

$B\cap D=\{(4,3)\}$

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

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

Directions: Given the events in the example above, find the following outcome sets.
1. $D\cap C$ and $D\cup C$
$D\cap C=\{(2, 5)\}$, $D\cup C=\{(1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (6,1), (5, 2), (4, 3), (3, 4), (1, 6)\}$
2. $A\cap D$ and $A\cup D$
$A\cap D=\{\emptyset\}$, $A\cup D=\{(6, 1), (5,2), (4,3), (3, 4), (2, 5), (1, 6), (3, 5)\}$
3. $B\cap C$ 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 2-4

Suppose events $A$, $B$, and $C$ are events of a particular scenario. Write the following using event notation.

1. 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.
2. None of the events occur.
Answer: $A^\prime\cap B^\prime \cap C^\prime$
3. Only A occurs.
Answer: $A\cap (B\cup C)^\prime$

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