Set theory for probability

Understanding the basics of set theory is a prerequisite for studying probability.

This lecture presents a concise introduction to set membership and inclusion, unions, intersections and complements. These are all concepts that are frequently used in the calculus of probabilities.

Table of contents

Sets
Set membership
Set inclusion
Union
Intersection
Complement
De Morgan's Laws
Solved exercises

Exercise 1
Exercise 2
Exercise 3

Applications to probability

Sets

A set is a collection of objects. Sets are usually denoted by a letter and the objects (or elements) belonging to a set are usually listed within curly brackets.

Example Denote by the letter the set of the natural numbers less than or equal to . Then, we can write

Example Denote by the letter the set of the first five letters of the alphabet. Then, we can write

Note that a set is an unordered collection of objects, i.e. the order in which the elements of a set are listed does not matter.

Example The two setsandare considered identical.

Sometimes a set is defined in terms of one or more properties satisfied by its elements. For example, the setcould be equivalently defined aswhich reads as follows: " is the set of all natural numbers such that is less than or equal to ", where the colon symbol () means "such that" and precedes a list of conditions that the elements of the set need to satisfy.

Example The set [eq7] is the set of all natural numbers such that divided by is also a natural number, that is,

Set membership

When an element belongs to a set , we writewhich reads " belongs to " or " is a member of ".

On the contrary, when an element does not belong to a set , we writewhich reads " does not belong to " or " is not a member of ".

Example Let the set be defined as follows:Then, for example,and

Set inclusion

If and are two sets and if every element of also belongs to , then we writewhich reads " is included in " orand we read " includes ". We also say that is a subset of .

Example The set is included in the setbecause all the elements of also belong to . Thus, we can write

When but is not the same as (i.e., there are elements of that do not belong to ), then we writewhich reads " is strictly included in " orWe also say that is a proper subset of .

Example Given the sets [eq21] we have that [eq22] but we cannot write

Union

Let and be two sets. Their union is the set of all elements that belong to at least one of them and it is denoted by

Example Define two sets and as follows: [eq25] Their union is

If $A_{1}$ , $A_{2}$ , ..., $A_{n}$ are sets, their union is the set of all elements that belong to at least one of them and it is denoted by [eq27]

Example Define three sets $A_{1}$ , $A_{2}$ and $A_{3}$ as follows: [eq28] Their union is [eq29]

Intersection

Let and be two sets. Their intersection is the set of all elements that belong to both of them and it is denoted by

Example Define two sets and as follows: [eq25] Their intersection is

If $A_{1}$ , $A_{2}$ , ..., $A_{n}$ are sets, their intersection is the set of all elements that belong to all of them and it is denoted by [eq33]

Example Define three sets $A_{1}$ , $A_{2}$ and $A_{3}$ as follows: [eq28] Their intersection is [eq35]

Complement

Complementation is another concept that is fundamental in probability theory.

Suppose that our attention is confined to sets that are all included in a larger set , called universal set. Let be one of these sets. The complement of is the set of all elements of that do not belong to and it is indicated by

Example Define the universal set as follows:and the two sets [eq38] The complements of and are [eq39]

Also note that, for any set , we have

De Morgan's Laws

De Morgan' Laws are [eq41] and can be extended to collections of more than two sets: [eq42]

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Define the following sets: [eq43] Find the following union: [eq44]

Solution

The union can be written asThe union of the three sets $A_{2}$ , $A_{3}$ and $A_{4}$ is the set of all elements that belong to at least one of them: [eq46]

Exercise 2

Given the sets defined in the previous exercise, find the following intersection: [eq47]

Solution

The intersection can be written asThe intersection of the four sets $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_{4}$ is the set of elements that are members of all the four sets: [eq49]

Exercise 3

Suppose that and are two subsets of a universal set and that [eq50] Find the following union:

Solution

By using De Morgan's laws, we obtain [eq52]

Applications to probability

Now that you are familiar with the basics of set theory, you can see how it is used in probability theory.

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How to cite

Please cite as:

Taboga, Marco (2021). "Set theory", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/mathematical-tools/set-theory.