24 2. FUNDAMENTAL RELIABILITY MATHEMATICS
An event refers to a single outcome or a group of outcomes of an experiment. For example, the
even number of rolling a die, that is, f2,4,6gis an event, which includes three possible outcomes.
e number 4 of rolling die is also an event, which has only one outcome. For another example,
that the yield strength is more than 32 ksi for a material tensile test is an event, which includes
infinite possible outcomes.
Sample space is defined as the event that includes all the possible outcomes of an experiment.
For example, the sample space of rolling a die consists of numbers 1, 2, 3, 4, 5, and 6. e
sample space of tossing a coin will consist of “head” and “tail.” e sample space of a material
tensile test experiment for yield strength will consist of the infinite possible outcomes, that is,
fyield strength 0g.
Mutually exclusive events are the events that cannot happen at the same time. For example, the
event of the tail and the event of the head in tossing a coin are mutually exclusive. e event of
a failed component and the event of a safe component are mutually exclusive events too because
a component cannot be a safe status or a failure status at the same time. For another example,
the event of a number less than 3 and the event of even number in rolling die are not mutually
exclusive events. If number 2 in an experiment of rolling a die happens, both events happen.
2.3 SET THEORY
e set theory can conveniently describe the operation on the outcomes or events of an exper-
iment and will help us smoothly to understand the concepts and related simple operations of
probability.
A set, denoted by a capital letter such as A, is a well-defined collection of objects so that for
any given object we can say whether or not it belongs to the set. A set is the equivalent term
of the event. In this book, the object means sample points or outcomes of an experiment. A
set can be expressed by braces containing the specified sample point or simple points. For
example, in rolling a die experiment, a set containing numbers 1 and 4 can be expressed as
A D
f
1; 4
g
, where the capital symbol A is the name of this set and
f
1; 4
g
represents the collec-
tion of sample point 1 and sample point 4. For an experiment of ultimate tensile strength S
u
,
B D
f
5 ksi < S
u
< 20 ksi
g
is a set. e set B includes the sample point of the ultimate tensile
strength is larger than 5 ksi and less than 20 ksi.
A universal set, denoted by Greek letter , is a collection of all possible sample points of the
experiment. For example, the universal set of rolling dice is D
f
1; 2; 3; 4; 5; 6
g
. For another
example, the universal set of the status of a component is D
f
safe; failure
g
.
An empty set or null set, denoted by the symbol ;, is a set containing no sample point of the
experiment. A null set can be expressed by ; D
f g
. e creation of a null set is mainly for the
set operations.
2.3. SET THEORY 25
e purpose of introducing some basic operations of the set is to help us to understand
some basic calculations of probability. We will discuss the following basic operations of the set.
Union of two sets. Union of two sets A and B, denoted A [ B, is the set of all objects that
belong to A, or B or both. e union operation can be graphically represented by the Venn
diagram in Figure 2.1.
A B
A∪B
Figure 2.1: Union of two sets A and B.
For example, in rolling a dice experiment, if the set A and B are: A D
f
1; 2
g
, and B D
f
5
g
,
the union of two sets A and B will be A [ B D
f
1; 2; 5
g
.
e intersection of two sets. e intersection of two sets A and B, denoted as A B, is the set
of all objects that belong to both set A and set B. e intersection of two sets can be graphically
represented by the Venn diagram in Figure 2.2.
A
A∩B
B
Figure 2.2: Intersection of two sets A and B.
For example, the intersection of f1; 2; 3g and f3; 4; 6g is the set f3g. e intersection of
f
1; 2; 3
g
and
f
5; 6
g
will be a null set
f g
because no object belongs to both sets
f
1; 2; 3
g
and
f
5; 6
g
.
e complement of a set. e complement of a set A denoted as A, is the set of all those objects
of the universal set which do not belong to A. e complement of a set A can be graphically
represented by the Venn diagram Figure 2.3.
For example, in rolling a dice experiment, the universal set is D
f
1; 2; 3; 4; 5; 6
g
. e
complement A of a set A D
f
1; 2
g
is A D
f
3; 4; 5; 6
g
. It is obvious that the union of a set A and
its complement set A will always be the universal set , that is, A [ A.
26 2. FUNDAMENTAL RELIABILITY MATHEMATICS
A
Ā
A
Figure 2.3: Complement of set A.
e following are several rules about the operations of union, interaction, and complement
of sets.
Commutative rule: e union and interaction of sets are commutative, that is,
A [B D B [ A (2.1)
A B D B A: (2.2)
Example 2.1
In rolling a dice experiment, sets A and B are A D
f
1; 2; 3; 4
g
and B D
f
1; 4; 5
g
. Use these two
sets to demonstrate Equations (2.1) and (2.2).
Solution:
According to the definition of the union of two sets, we have:
A [B D
f
1; 2; 3; 4
g
[
f
1; 4; 5
g
D
f
1; 2; 3; 4; 5
g
B [ A D
f
1; 4; 5
g
[
f
1; 2; 3; 4
g
D
f
1; 2; 3; 4; 5
g
:
erefore, A [ B D B [ A.
According to the definition of the intersection of two sets, we have:
A B D
f
1; 2; 3; 4
g
f
1; 4; 5
g
D
f
1; 4
g
B A D
f
1; 4; 5
g
f
1; 2; 3; 4
g
D
f
1; 4
g
:
erefore, A B D B A D
f
1; 4
g
.
Associative rule: e union and intersection of sets are associative, that is,
.
A [B
/
[ C D A [
.
B [ C
/
(2.3)
.
A B
/
C D A
.
B C
/
: (2.4)
2.3. SET THEORY 27
Example 2.2
In rolling a dice experiment, sets A, B, and C are A D
f
1; 4
g
, B D
f
2; 4; 5
g
, and C D
f
3
g
. Use
these three sets to demonstrate Equations (2.3) and (2.4).
Solution:
According to the definition of the union of two sets, we have:
.
A [B
/
[ C D
.
f
1; 4
g
[
f
2; 4; 5;
g
/
[
f
3
g
D
f
1; 2; 4; 5;
g
[
f
3
g
D
f
1; 2; 3; 4; 5
g
A [
.
B [ C
/
D
f
1; 4
g
[
.
f
2; 4; 5;
g
[
f
3
g
/
D
f
1; 4
g
[
f
2; 3; 4; 5;
g
D
f
1; 2; 3; 4; 5
g
:
erefore,
.
A [B
/
[ C D A [
.
B [ C
/
D
f
1; 2; 3; 4; 5
g
.
According to the definition of the intersection of two sets, we have:
.
A B
/
C D
.
f
1; 4
g
f
2; 4; 5;
g
/
f
3
g
D
f
4
g
f
3
g
D ;
A
.
B C
/
D
f
1; 4
g
.
f
2; 4; 5;
g
f
3
g
/
D
f
1; 4
g
; D ;:
erefore,
.
A B
/
C D A
.
B C
/
D ;.
Distributive rule: e union and intersection of sets are distributive.
.
A [B
/
C D
.
A C
/
[
.
B C
/
(2.5)
.
A B
/
[ C D
.
A [C
/
.
B [ C
/
: (2.6)
Example 2.3
In rolling a dice experiment, sets A, B, and C are A D
f
2; 3
g
, B D
f
1; 4; 5
g
, and C D
f
1; 3; 6
g
.
Use these three sets to demonstrate Equations (2.5) and (2.6).
Solution:
According to the definitions of union and intersection of two sets, we have:
.
A [B
/
C D
.
f
2; 3
g
[
f
1; 4; 5
g
/
f
1; 3; 6
g
D
f
1; 2; 3; 4; 5
g
f
1; 3; 6
g
D
f
1; 3
g
.
A C
/
[
.
B C
/
D
.
f
2; 3
g
f
1; 3; 6
g
/
[
.
f
1; 4; 5
g
f
1; 3; 6
g
/
D
f
3
g
[
f
1
g
D
f
1; 3
g
:
So,
.
A [B
/
C D
.
A C
/
[
.
B C
/
D
f
1; 3
g
.
According to the definitions of union and intersection of two sets, we have:
.
A B
/
[ C D
.
f
2; 3
g
f
1; 4; 5
g
/
[
f
1; 3; 6
g
D ; [
f
1; 3; 6
g
D
f
1; 3; 6
g
.
A [C
/
.
B [ C
/
D
.
f
2; 3
g
[
f
1; 3; 6
g
/
.
f
1; 4; 5
g
[
f
1; 3; 6
g
/
D
f
1; 2; 3; 6
g
f
1; 3; 4; 5; 6
g
D
f
1; 3; 6
g
:
erefore,
.
A B
/
[ C D
.
A [C
/
.
B [ C
/
D
f
1; 3; 6
g
.
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