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.
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 setis the set of all natural numbers such that divided by is also a natural number, that is,
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
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 we have thatbut we cannot write
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:Their union is
If , , ..., are sets, their union is the set of all elements that belong to at least one of them and it is denoted by
Example Define three sets , and as follows:Their union is
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:Their intersection is
If , , ..., are sets, their intersection is the set of all elements that belong to all of them and it is denoted by
Example Define three sets , and as follows:Their intersection is
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 setsThe complements of and are
Also note that, for any set , we have
De Morgan' Laws areand can be extended to collections of more than two sets:
Below you can find some exercises with explained solutions.
Define the following sets:Find the following union:
The union can be written asThe union of the three sets , and is the set of all elements that belong to at least one of them:
Given the sets defined in the previous exercise, find the following intersection:
The intersection can be written asThe intersection of the four sets , , and is the set of elements that are members of all the four sets:
Suppose that and are two subsets of a universal set and thatFind the following union:
By using De Morgan's laws, we obtain
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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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.
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