1
SETS, SAMPLE SPACES AND EVENTS
Mathematical Models
What is a mathematical model?
Application
Observation
If the
• representation is honest,
• theory is correct, and
• interpretation is astute,
then the result may, indeed, be the solution to the problem.
Deterministic models
For specified values of an experiment’s input parameters, the result is a known
constant.
Example: V = IR
Established experimentally by George Simon Ohm in 18271
1
To place this date in perspective, University at Buffalo was founded in 1846.
1
2
Sets
Nondeterministic models
Also called probabilistic models or stochastic models.
Examples:
• the number of α-particles emitted from a piece of radioactive material in one
minute.
• the outcome of a single toss of a fair die.
• the number of students who will earn an A in EAS305.
Sets
Basic definitions
We will need to use set theory to describe possible outcomes and scenarios in our
model of uncertainty.
Definition 1.1. A set is a collection of elements.
Notation: Sets are denoted by capital letters (A, B, Ω, . . .) while their
elements are denoted by small letters (a, b, ω, . . .). If x is an element of
a set A, we write x ∈ A. If x is not an element of A, we write x 6∈ A.
A set can be described
• by listing the set’s elements
A = {1, 2, 3, 4}
• by describing the set in words
“A is the set of all real numbers between 0 and 1, inclusive.”
• by using the notation {ω : specification for ω}
A = {x : 0 ≤ x ≤ 1}
or
A = {0 ≤ x ≤ 1}
Sets
3
There are two sets of special importance:
Definition 1.2. The universe is the set containing all points under consideration
and is denoted by Ω.
Question: What is the difference (if any) between Ø and {Ø}?
Definition 1.3. The empty set (the set containing no elements) is denoted by Ø.
Definition 1.4. If every point of set A belongs to set B , then we say that A is a
subset of B (B is a superset of A). We write
A⊆B
B⊇A
Now we have a way to say A is identical to B . . .
Definition 1.5. A = B if and only if A ⊆ B and B ⊆ A.
Example: Let Ω = R, the set of all real numbers. Define the sets
A = {x : x2 + x − 2 = 0}
B = {x : (x − 3)(x2 + x − 2) = 0}
C = {−2, 1, 3}
Then A ⊆ B and B = C .
Definition 1.6. A is a proper subset of B if A ⊆ B and A 6= B . We write
A⊆
/ B.
Note: Some authors use the symbol ⊂ for subset and other use ⊂ for proper subset.
We avoid the ambiguity here by using ⊆ for subset and ⊆
/ for proper subset.
Venn diagrams
Ω
B
A
4
Sets
Set operations
Complementation
Ac = {ω ∈ Ω : ω 6∈ A}
Example: Here is the region representing Ac in a Venn diagram:
Ω
Ac
A
Union
A ∪ B = {ω ∈ Ω : ω ∈ A or ω ∈ B (or both)}
Example: Consider the following Venn diagram:
Ω
A
then the region representing A ∪ B is:
B
Sets
5
Intersection
A ∩ B = {ω ∈ Ω : ω ∈ A and ω ∈ B}
Example: Using the same Venn diagram as above the region representing A ∩ B
is given by:
Note that some textbooks use the notation AB instead of A ∩ B to denote the
intersection of two sets.
Definition 1.7. Two sets A and B are disjoint (or mutually exclusive) if A∩B =
Ø.
These basic operations can be extended to any finite number of sets.
A ∪ B ∪ C = A ∪ (B ∪ C) = (A ∪ B) ∪ C
and
A ∩ B ∩ C = A ∩ (B ∩ C) = (A ∩ B) ∩ C
You can show that
(a) A ∪ B = B ∪ A
(b) A ∩ B = B ∩ A
(c) A ∪ (B ∪ C) = (A ∪ B) ∪ C
(d) A ∩ (B ∩ C) = (A ∩ B) ∩ C
Note: (a) and (b) are the commutative laws
Note: (c) and (d) are the associative laws
6
Sets
Set identities
(e) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(f) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
(g) A ∩ Ø = Ø
(h) A ∪ Ø = A
(i) (A ∪ B)c = Ac ∩ B c
(j) (A ∩ B)c = Ac ∪ B c
(k) (Ac )c = A
Note: (e) and (f) are called the Distributive Laws
Note: (i) and (j) are called DeMorgan’s Laws
Proving statements about sets
Venn diagrams can only illustrate set operations and basic results. You cannot
prove a true mathematical statement regarding sets using Venn diagrams. But they
can be used to disprove a false mathematical statement.
Here is an example of a simple set theorem and its proof.
Theorem 1.1. If A ⊆ B and B ⊆ C , then A ⊆ C .
Proof.
(1)
(2)
(3)
(4)
Discussion
If x ∈ A then x ∈ B
Since x ∈ B then x ∈ C
If x ∈ A then x ∈ C
Therefore A ⊆ C
Reasons
Definition of A ⊆ B
Definition of B ⊆ C
By statements (1) and (2)
Definition of A ⊆ C
Note: A proof is often concluded with the symbol
letters QED.
(as in the above proof) or the
Many mathematicians prefer to write proofs in paragraph form. For example, the
proof of Theorem 1.1 would become:
Sets
7
Proof. Choose an x ∈ A. Since A ⊆ B , x ∈ A implies x ∈ B , from the definition
of subset. Furthermore, since B ⊆ C , x ∈ B implies x ∈ C . Therefore, every
element x ∈ A is also an element of C . Hence A ⊆ C .
A proof of the distributive law for sets
Here is a proof of one of the distributive laws for sets. The proof is from a textbook
on set theory by Flora Dinkines2
Theorem 1.2. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Proof.
(1)
(2)
(3)
(4)
(5)
Discussion
If x ∈ A ∩ (B ∪ C), then
x ∈ A and x ∈ B ∪ C .
Since x ∈ B ∪ C , then
x ∈ B or x ∈ C .
Therefore x ∈ A and x ∈ B ,
or x ∈ A and x ∈ C .
Hence x ∈ A ∩ B or
x ∈ A ∩ C.
x ∈ (A ∩ B) ∪ (A ∩ C).
Reasons
Definition of ∩.
Definition of ∪.
By (1) and (2).
Definition of ∩.
Definition of ∪.
Therefore every element of A ∩ (B ∪ C) is also an element of (A ∩ B) ∪ (A ∩ C),
giving us
A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C).
We now prove the inclusion in the opposite direction:
2
Dinkines, F., Elementary Theory of Sets, Meredith Publishing, 1964.
8
Sets
(6)
(7)
(8)
(9)
(10)
(11)
If y ∈ (A ∩ B) ∪ (A ∩ C),
then y ∈ (A ∩ B) or
y ∈ (A ∩ C).
If y ∈ A ∩ B then
y ∈ A and y ∈ B .
If y ∈ A ∩ C then
y ∈ A and y ∈ C .
In either case (7) or (8),
y ∈ A and y is an
element of one of the
sets B or C .
Therefore y ∈ A and
y ∈ B ∪ C.
Therefore y ∈ A ∩ (B ∪ C).
Definition of ∪.
Definition of ∩.
Same as (7).
Statements (7) and (8)
Since y is in B or C ,
it is in the union.
Definition of ∩.
Therefore every element of (A ∩ B) ∪ (A ∩ C) is also an element of A ∩ (B ∪ C),
giving us
A ∩ (B ∪ C) ⊇ (A ∩ B) ∪ (A ∩ C).
Therefore, using the definition of set equality, we have
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Elementary sets
Let Ω be a universe with three subsets A, B , and C . Consider the following subsets
of Ω:
A B C
0 0 0
Ac ∩ B c ∩ C c = S0
0 0 1
Ac ∩ B c ∩ C
= S1
0 1 0
Ac ∩ B ∩ C c
= S2
0 1 1
Ac ∩ B ∩ C
= S3
1 0 0
A ∩ Bc ∩ C c
= S4
1 0 1
A ∩ Bc ∩ C
= S5
1 1 0
A ∩ B ∩ Cc
= S6
1 1 1
A∩B∩C
= S7
Sets
9
Ω
B
A
A Bc Cc
A B Cc
A B C
A Bc C
Ac B Cc
Ac B C
C
Ac Bc C
Ac Bc Cc
Figure 1.1: An example of elementary sets for three subsets
For example, the binary sequence (0,1,1) corresponds to S3 = Ac ∩ B ∩ C . This
is the subset of elements that are
0:
1:
1:
not in A
in B
in C
and
and
Figure 1.1 illustrates all of the 23 = 8 elementary sets for A, B and C using a Venn
diagram.
The result of any set operations involving A, B , and C can be represented as a
union of some of the elementary sets Si .
For example
(A ∪ B) ∩ C = S3 ∪ S5 ∪ S7
Cardinality of sets
The cardinality of a set A (denoted by |A|) is the number of elements in A.
For some (but not all) experiments, we have
• |Ω| is finite.
10
Sets
• Each of the outcomes of the experiment are equally likely.
In such cases, it is important to be able to enumerate (i.e., count) the number of
elements of subsets of Ω.
Sample spaces and events
Using set notation, we can now introduce the first components of our probability
model.
Definition 1.8. A sample space, Ω, is the set of all possible outcomes of an
experiment.
Definition 1.9. An event is any subset of a sample space.
Example: Consider the experiment of rolling a die once.
3 What should be Ω for this experiment?
Ω=
o
n
If you thought the answer was Ω = {1, 2, 3, 4, 5, 6}, then you
just (re)invented the concept of a random variable. This will be
discussed in a later chapter.
3 Identify the subsets of Ω that represent the following events:
1. The die turns up even.
n
o
A=
2. The die turns up less than 5.
n
o
B=
3. The die turns up 6.
n
o
C=
Example: Consider the experiment of burning a light bulb until it burns out. The
outcome of the experiment is the life length of the bulb.
3 What should be Ω for this experiment?
Ω = {x ∈ R : x ≥ 0}
Sets
11
For the lack of a better representation, we resorted to assigning a
nonnegative real number to each possible outcome of the experiment.
3 Identify the subsets of Ω that represent the following events:
1. The bulb burns out within one and one-half hours.
A = {x ∈ R : 0 ≤ x < 1.5}
2. The bulb lasts at least 5 hours.
B = {x ∈ R : x ≥ 5}
3. The bulb lasts exactly 24 hours.
C = {24}
Sets of events
It is sometimes useful to be able to talk about the set of all possible events F. The
set F is very different than the set of all possible outcomes, Ω. Since an event is a
set, the set of all events is really a set of sets. Each element of F is a subset of Ω.
For example, suppose we toss a two-sided coin twice. The sample space is
Ω = {HH, HT, T H, T T }.
If you were asked to list all of the possible events for this experiment, you would
need to list all possible subsets of Ω, namely
A1 = {{HH}}
A2 = {{HT }}
A3 = {{T H}}
A4 = {{T T }}
A5 = {{HH}, {HT }}
A6 = {{HH}, {T H}}
A7 = {{HH}, {T T }}
A8 = {{HT }, {T H}}
12
Self-Test Exercises for Chapter 1
A9 = {{HT }, {T T }}
A10 = {{T H}, {T T }}
A11 = {{HH}, {HT }, {T H}}
A12 = {{HH}, {HT }, {T T }}
A13 = {{HH}, {T H}, {T T }}
A14 = {{HT }, {T H}, {T T }}
A15 = {{HH}, {HT }, {T H}, {T T }} = Ω
A16 = {} = Ø
In this case
F = {A1 , A2 , . . . , A16 }
Definition 1.10. For a given set A, the set of all subsets of A is called the power
set of A, and is denoted by 2A .
Theorem 1.3. If A is a finite set, then |2A | = 2|A|
Proof. Left to reader.
If Ω is a sample space, then we often use the set of all possible events F = 2Ω .
When Ω contains a finite number of outcomes, this often works well. However, for
some experiments, Ω contains an infinite number of outcomes (either countable or
uncountable) and F = 2Ω is much too large. In those cases, we select a smaller
collection of sets, F, to represent those events we actually need to consider. How
to properly select a smaller F, is found in many graduate level probability courses.
Self-Test Exercises for Chapter 1
For each of the following multiple-choice questions, choose the best response
among those provided. Answers can be found in Appendix B.
S1.1 The event that corresponds to the statement, "at least one of the events A, B
and C occurs," is
(A) A ∪ B ∪ C
(B) A ∩ B ∩ C
(C) (A ∩ B ∩ C)c
(D) (A ∪ B ∪ C)c