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STATLECT > Mathematical tools

Set theory

This lecture introduces the basics of set theory.

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, i.e.:[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:[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:

  1. Exercise set 1 (simple exercises on unions, intersections and complements)

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