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Set theory

This lecture introduces the basics of set theory.

Table of Contents

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 $S$ the set of the natural numbers less than or equal to $5$. Then, we can write[eq1]

Example Denote by the letter A the set of the first five letters of the alphabet. Then, we can write[eq2]

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 sets[eq3]and[eq4]are considered identical.

Sometimes a set is defined in terms of one or more properties satisfied by its elements. For example, the set[eq5]could be equivalently defined as[eq6]which reads as follows: "$S$ is the set of all natural numbers n such that n is less than or equal to $5$", 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 n such that n divided by $4$ is also a natural number, that is,[eq8]

Set membership

When an element a belongs to a set A, we write[eq9]which reads "a belongs to A" or "a is a member of A".

On the contrary, when an element a does not belong to a set A, we write[eq10]which reads "a does not belong to A" or "a is not a member of A".

Example Let the set $S$ be defined as follows:[eq11]Then, for example,[eq12]and[eq13]

Set inclusion

If A and $B$ are two sets and if every element of A also belongs to $B$, then we write[eq14]which reads "A is included in $B$" or[eq15]and we read "$B$ includes A". We also say that A is a subset of $B$.

Example The set [eq16]is included in the set[eq17]because all the elements of A also belong to $B$. Thus, we can write[eq18]

When $Asubseteq B$ but A is not the same as $B$ (i.e., there are elements of $B$ that do not belong to A), then we write[eq19]which reads "A is strictly included in $B$" or[eq20]We also say that A is a proper subset of $B$.

Example Given the sets [eq21]we have that[eq22]but we cannot write[eq23]

Union

Let A and $B$ be two sets. Their union is the set of all elements that belong to at least one of them and it is denoted by[eq24]

Example Define two sets A and $B$ as follows:[eq25]Their union is[eq26]

If $A_{1}$, $A_{2}$, ..., $A_{n}$ are n 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 A and $B$ be two sets. Their intersection is the set of all elements that belong to both of them and it is denoted by[eq30]

Example Define two sets A and $B$ as follows:[eq25]Their intersection is[eq32]

If $A_{1}$, $A_{2}$, ..., $A_{n}$ are n 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

Suppose that our attention is confined to sets that are all included in a larger set Omega, called universal set. Let A be one of these sets. The complement of A is the set of all elements of Omega that do not belong to A and it is indicated by[eq36]

Example Define the universal set Omega as follows:[eq37]and the two sets[eq38]The complements of A and $B$ are[eq39]

Also note that, for any set A, we have[eq40]

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 as[eq45]The 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 as[eq48]The 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 A and $B$ are two subsets of a universal set Omega and that[eq50]Find the following union:[eq51]

Solution

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

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