APPENDIX A: SET THEORY Bain and Engelhardt pp 587-593

1 [Appendix A]



APPENDIX A: SET THEORY
Bain and Engelhardt pp 587-593


A.1 SET THEORY
The theory of mathematical statistics is based on probability theory and probability theory
in turn requires a sound knowledge of set theory.
A set is simply a collection of distinguishable elements. For every possible element we
must be able to determine whether that element belongs to the set or whether it does not
belong to the set. If an element e belongs to the set A we say that e ∈ A. If the element e
does not belong to the set A we say that e ø A.
It is the normal practice to indicate sets by capital letters and to indicate the elements of
sets by small letters. There are two ways used to indicate sets and their elements .
1. All the elements of the set are written down and enclosed in two curly brackets e.g. if
A = {1, 2, 3} then A is the set which consists of the three elements 1, 2 and 3. Note
that the order in which the elements are written down does not make any difference.
In this example 2 ∈ A while 5 ø A.
2. The notation A = {x | p(x)} is used to indicate that the set A consists of all elements
for which the statement p is true. Hence x ∈ A if and only if the statement p(x) is
true. Therefore, if A = {x | x a real number such that 0 < x < 1} then A consists of
all real numbers greater than 0 and less than 1.
The set of all possible elements which is part of a certain discussion is called the space
and is usually indicated by the symbol Ω. For all possible elements e it is true that e ∈ Ω.
The set of all real numbers is indicated by R.


oDefinition A.1.1 :
We say that A is a subset of B if all elements of A are also elements of B and is indicated
by A < B. Hence A < B if and only if
e ∈ A → e ∈ B for all e ∈ A.
For any set A it is true that A < Ω since all the elements of A are elements of the set of
all possible elements i.e. Ω.
The empty set is the set that does not have any elements and is indicated by ф. Note that
ф < A for any possible set A since the statement that if e ∈ ф then e ∈ A is true sincethere
are no elements e such that e ∈ ф .

,2 [Appendix A]


oDefinition A.1.2 :
The union of two sets A and B is the set of all elements e such that e ∈ A or e ∈ B and
is indicated by A ∪ B. Hence
A ∪ B = {e | e ∈ A or e ∈ B}.
Put differently, A ∪ B consists of all elements which belongs to at least one of the sets A
or B i.e.
A ∪ B = {e | e an element of at least one of A or B }


yEXAMPLES :
Example A.1.1
Suppose that Ω = R and that A = {1, 2, 3}, B = {3, 4, 5, 6} and C = {4, 7}.
Then A ∪ B = {1, 2, 3, 4, 5, 6}, A ∪ C = {1, 2, 3, 4, 7} and B ∪ C = {3, 4, 5, 6, 7}.
Example A.1.2
Suppose that Ω = R and that D = {x | 0 < x < 2}, E = {x | 1 < x < 4} and
F = {x | 1. 5 < x < 7}.
Then D ∪ E = {x | 0 < x < 4}, D ∪ F = {x | 0 < x < 7} and E ∪ F = {x | 1 < x < 7}.


oDefinition A.1.3 :
The intersection of two sets A and B is the set of all elements e such that e ∈ A and
e ∈ B and is indicated by A ß B. Hence
A ß B = {e | e ∈ A and e ∈ B}.
The set A ß B therefore consists of all elements e which belongs to both A and B.


oDefinition A.1.4 :
The complement of a set A is the set of all elements e such that e is not an element of A
and is indicated by –A. Hence
– A = {e | e ∈ Ω and e ø A}.


yEXAMPLE A.1.3 :
Suppose that Ω = R and that A = {x | x ≤ 0} and B = {x | x = 2}.
Then – A = {x | x > 0} and – B = {x | x m 2}.

,3 [Appendix A]


♦ Theorem A.1.1 :
–Ω = ф
Proof : –Ω = {e | e ø Ω}
= ф.
Since there are no elements e for which it is true that e is not an element of Ω. ☺


♦ Theorem A.1.2 :
–(A ∪ B) = (–A) ß (–B).
Proof :
–(A ∪ B) = – e | e ∈ A or e ∈ B
= e | it is not true that e ∈ A or e ∈ B
= e | e ø A and e ø B
= e | e ∈ –A and e ∈ –B
= (–A) ß (–B). ☺


♦ Theorem A.1.3 :
–(A ß B) = (–A) ∪ (–B).
Proof :
–(A ß B) = – e | e ∈ A and e ∈ B
= e | it is not true that e ∈ A and e ∈ B
= e | e is not an element of A or e is not an element of B
= e | e ø A or e ø B
= e | e ∈ –A or e ∈ –B
= (–A) ∪ (–B). ☺


Note: Theorems A.1.2 and A.1.3 are known as De Morgan’s laws.


oDefinition A.1.5 :
Two sets A and B are disjoint if there are no elements which belong to both A and B i.e.
A ß B = ф.

, 4 [Appendix A]


The definitions of union and intersection can be extended to more than two sets or even
an infinite number of sets as follows :
A∪B∪C = e | e ∈ A or e ∈ B or e ∈ C
∞
∪ Ai = e | e ∈ Ai for at least one i = 1, 2, 3, …
i=1

AßBßC = e | e ∈ A and e ∈ B and e ∈ C .
∞
and ß Ai = e | e ∈ Ai for all i = 1, 2, 3, …
i=1




yEXAMPLE A.1.4 :
Suppose that Ω = R and that Ai = x|0 < x < i and Bi = x | 0 < x < 1/i .
Then
∞ ∞
∪ Ai = x | x > 0 and ß Ai = x | 0 < x < 1 and
i=1 i=1
∞ ∞

∪ Bi = x|0 < x < 1 and ß Bi = x|0 < x < 0 = ф.
i=1 i=1




♦ Theorem A.1.4 :
∞ ∞
– ∪ Ai = ß – Ai .
i=1 i=1
Proof :
∞

– ∪ Ai = – e | e ∈ Ai for at least one i = 1, 2, 3, …
i=1
= e | it is not true that e ∈ Ai for at least one i = 1, 2, 3, …
= e | e does not belong to one of Ai for i = 1, 2, 3, …
= e | e ø Ai for all i = 1, 2, 3, …
= e | e ∈ –Ai for all i = 1, 2, 3, …
∞
= ß – Ai . ☺
i=1