The review of set theory contained herein adopts a naive point of view. We assume that the meaning of a set as a collection of objects is intuitively clear. A rigorous analysis of the concept belongs to the foundations of mathematics and mathematical logic. Although we shall not initiate a study of these fields, the rules we follow in dealing with sets are derived from them. A set is a collection of objects, which are the elements of the set.
If an element
belongs to a set
, we express this fact by writing
. If
does not belong to
, we write
. We use the equality symbol to denote logical identity. For instance,
means that
are symbols denoting the same object. Similarly, the equation
states that
are two symbols for the same set. In particular, the sets
contain precisely the same elements. If
are different objects then we write
. Also, we can express the fact that
are different sets by writing
.
A set
is a subset of
if every element of
is also contained in
. We express this relation by writing
. Note that this definition does not require
to be different from
. In fact,
if and only if
and
. If
and
is different from
, then
is a proper subset of
and we write
.
There are many ways to specify a set. If the set contains only a few elements, one can simply list the objects in the set;

The content of a set can also be enumerated whenever
has a countable number of elements,

Usually, the way to specify a set is to take some collection
of objects and some property that elements of
may or may not possess, and to form the set consisting of all elements of
having that property. For example, starting with the integers
, we can form the subset of S consisting of all even numbers

More generally, we denote the set of all elements that have a certain property
by

The braces are to be read as the words "the set of" whereas the symbol | stands for the words "such that."
It is convenient to introduce two special sets. The empty set, denoted by
, is a set that contains no elements. The universal set is the collection of all objects of interest in a particular context, and it is denoted by
. Once a universal set
is specified, we need only consider sets that are subsets of
. In the context of probability,
is often called the sample space.
The complement of a set
, with respect to the universal set
, is the collection of all objects in
that do not belong to
,

We note that
.
Probability theory makes extensive use of elementary set operations. Below, we review the ideas of set theory, and establish the basic terminology and notation. Consider two sets
.
The union of sets
is the collection of all elements that belong to
or
(or both), and it is denoted by
. Formally, we define the union of these two sets by

The intersection of sets
is the collection of all elements that are common to both
and
. It is denoted by
, and it can be expressed mathematically as

When
have no elements in common, we write
. We also express this fact by saying that
are disjoint. More generally, a collection of sets is said to be disjoint if no two sets have a common element. A collection of sets is said to form a partition of
if the sets in the collection are disjoint and their union is
.
The difference of two sets, denoted by
or
, is defined as the set consisting of those elements of
that are not in
,

This set is sometimes called the complement of
relative to
, or the complement of
in
.
We have already looked at the definition of the union and the intersection of two sets. We can also form the union or the intersection of arbitrarily many sets. This is defined in the obvious way,


The index set I can be finite or even infinite.
Given a collection of sets, it is possible to form new ones by applying elementary set operations to them. As in algebra, one uses parentheses to indicate precedence. For instance,
denotes the union of two sets
, while
represents the intersection of two sets
.The sets thus formed are quite different.
Sometimes different combinations of operations lead to the same set. For instance, we have the two distributive laws


Two particularly useful equivalent combinations of operations are given by De Morgan's laws, which state that


These two laws can be generalized to


when multiple sets are involved. To establish the first equality, suppose that
belongs to
. Then
is not contained in
.
That is,
is not an element of
for any
. This implies that
belongs to
for all
, and therefore
. We have shown that
. The converse inclusion is obtained by reversing the above argument. The second law can be obtained in a similar fashion.
There is yet another way to create new sets form existing ones. It involves the notion of an ordered pair of objects. Given sets
, the cartesian product
is the set of all ordered pairs
for which
is an element of
and
is an element of
,

Summary of Probability & Set Theory[edit | edit source]
![{\displaystyle A=P(A)\cdot \sum [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec52c5fd0b4fcc1ed1e36c2efa94297a3da41f8)

[*if and only if
and
are mutually exclusive]
[*if and only if
and
are independent]
[*conditional]
- Basic Set Theory
UNION: combined area of set
, known as
. The set of all items which are members of either
or
- Union of
are added together
- Some basic properties of unions:




, where 
, if and only if 
INTERSECTION: area where both
overlap, known as
. It represents which elements the two sets
have in common
- If
, then A and B are said to be DISJOINT.
- Some basic properties of intersections:





, if and only if 
- UNIVERSAL SET: space of all things possible, which contains ALL of the elements or elementary events.
is called the absolute complement of 
Complement (set): 2 sets can be subtracted. The relative complement (set theoretic difference of
and
). Denoted by
(or
) is the set of all elements which are members of
, but not members of
Some basic properties of complements (
):




, and 

- Summary
- Intersection
--> AND – both events occur together at the same time
- Union
--> OR – everything about both events, 
- Complement
--> NOT
– everything else except
(or the event in question)
(sample space)
(impossible event)
Union and Intersection are:
- Commutative


- Associative


- Distributive

