Introduction to Set Theory

Math, Set Theory / By /
marbles

Contents

  1. Introduction

  2. What Is a set?

  3. Operations with Sets

  4. What Are Spaces?

Introduction

Set theory is a foundational mathematical discipline and understanding its fundamentals is necessary in order to explore probability and statistics.

Generally, set theories are concerned with collections called sets, which contain unique objects called elements or members. Most commonly, the elements in sets must be sets themselves. There are many different theories of sets. The most common and applicable to general mathematics are the theories built on the principle of extensionality, which tells us sets are only defined by their membership. This introduction covers material that is generally true across all extensionality based theories (unless otherwise noted).

A Unique Collection of Things

collection of leaves

A set is a collection of unique elements. An element can exist in one, or more sets, in which case that element is said to be a member of those sets.

Sets are most often represented by curly braces. A symbol appearing next to the braces is the identity for the set itself. See an example below.

\[\large{
\begin{array}{cc}
\stackrel{\text{x is a set.}\\}{x:\{1,2,3\}} && \stackrel{\text{1 is a member of x.}\\}{1\in x}
\end{array}
}\]

Extensionality

A principle around which most set theories are built is called extionsionality: sets are defined only by their membership (i.e. if 2 sets have exactly and only the same elements, then they are the same set). Extensionality also implies that while other properties, such as the order of the elements, can be defined they cannot change the definition of sets: {A,B,C} is the same set as {C,A,B}.

Distinguishable Things

By implication, if sets hold unique collections, then the members must be distinguishable from one another.

Imagine a bag of 4 marbles, perfectly round, exactly the same circumference, all solid white, and indistinguishable from one another. Does this bag of marbles constitute a set? No! A set is a collection of unique elements and since we cannot distinguish the marbles we cannot determine their uniqueness and cannot define a set using them. If we cannot check the truth of the statement A = B, then {A,B} is NOT a set.

However, if we know A = B is true, then the set {A,B} is a set with only the unique elements respected in any given context (i.e. essentially treat {A,B} as if it were {A} or {B} if it is known A=B).

Any Type of Thing

Strictly speaking a set can only hold unique elements defined in the set theory, which generally means sets can only contain sets. However, being unique makes the elements of sets identities, which means they can be used to represent other objects.

The most obvious application for math is to use sets to represent numbers. See an example of the most common method (the Von Neumann construction) for natural numbers represented as sets.

\[\large{
\begin{align}
\{\} &\equiv 0 && \small\text{(0 is equivalent to the empty set)}\\ \\
\{0\} &\equiv 1 && \small\text{(1 is the set containing 0)}\\ \\
\{0,1\} &\equiv 2 && \small\text{(2 is the set containing 0 and 1)} \\ \\
\{0,1,2\} &\equiv 3 && \small\text{(2 is the set containing 0, 1, and 2)}
\end{align}
}\]

Sets can also be used to represent any type of mathematical object (numbers, sets, vectors, shapes, trees, functions, etc.) or non-mathematical object (people, dollars, cars, houses, etc).

Special Sets

short bus

The previous section defined sets in general. This section will define some specific sets and categories of sets.

The Empty Set

The empty or null set is the set containing no elements. Below are two of the most common ways of expressing the empty set.

\[\large{\stackrel{\text{The Empty Set}\\}{\emptyset:\{\}\quad\text{or}\quad\Lambda:\{\}}}\]

The Singleton

A singleton is a set containing a single element: {A}.

Do not confuse the empty set for a singleton. ∅ is a set containing 0 elements, not 1 element. {∅} is the set containing the empty set and it is a singleton.

The Ordered Pair

An ordered pair is a set with a pair of elements and order defined for them. There is a lot of variation in the definition of ordered pair between different set theories. The exact construction of ordered pair can greatly impact working in a specific theory. However, for the purposes of elementary introduction all the definitions are roughly equivalent.

The most common definition is the Kuratowski ordered pair, which is a set containing two sets, which contain the inputs to the ordered pair. The set containing the 2nd (as read left to right) input variable to the ordered pair is the second set and the other is the first.

\[\large{
\stackrel{\text{Kuratowski Ordered Pair}\\}{\langle a,b \rangle = \Big\{\stackrel{\text{1st}}{\{a\}},\stackrel{\text{2nd}}{\{a,b\}}\Big\}}
}\]

Number Systems

Sets can also be defined as containers for specific number systems. By convention these types of sets are represented by a specific bold face letter.

  • The Set Containing All Natural aka “Countable” Numbers: \(\mathbb{N}\)
  • The Set Containing All Integers: \(\mathbb{Z}\)
  • The Set Containing All Real Numbers: \(\mathbb{R}\)
  • The Set Set Containing All Complex Numbers: \(\mathbb{C}\)

Universes

universe contained in a marble

Universe is an informal term for a set containing all the elements under consideration in a certain context. For example, “given universe U,” simply means given a set U, that for this context we will consider as the entire universe of elements (i.e. we assume no elements exist except those in the universe).

The universal set, usually represented \(\text{V}\), is the set of all sets including itself. It is the largest possible universe of sets. It is not an uncommon concept, but not all set theories have a universal set (in fact, the most common and widely used theory of sets, Zermelo-Frankel, has no universal set). For the purpose of this material the universal set is not necessary, but you might run into references to it if you are searching other sources.

Common Set Operations

This section covers the most common operations that can be performed on sets (i.e. each of these operations accepts at least one variable that is a set or outputs a set).

Venn-Diagram

Let x and y be variables, φ be a formula, S:{1,2,3,4,5,6} be a universe, and A:{1,2,3} and B:{3,4,5} be sets:

  1. Subset/Superset: A ⊆ S or S ⊇ A
    • A is a subset of S or S is a superset of A.
    • A subset can be made from a superset by removing some number of elements from the superset.
    • A superset can be made from a subset by adding some number of elements to the subset.
    • The empty set is a subset of every set including itself (removing 0 elements from \(\Lambda\) equals \(\Lambda\)).
    • The universal set (if it exists) is a superset of every set including itself (adding 0 elements to \(\text{V}\) equals \(\text{V}\)).

  2. Proper Subset/Superset: A ⊂ S or S ⊃ A
    • A is a proper subset of S or S is a proper superset of A.
    • A proper sub/super set has fewer or more elements respectively than the set it is related to.
    • Every set is a sub and super set of itself (S ⊆ S and S ⊇ S), but not a proper sub or super set of itself.

  3. Powerset: \(\mathbb{P}\)(A) = {∅,{1},{2},{3},{1,2},{1,3},{2,3},A}
    • The powerset of A is the set of all subsets of A (including itself and the empty set).

  4. Membership: x ∈ y
    • x is a member of y or x is an element in y.
    • 1 ∈ A, {1} ∉ A, 1 ∈ S, A ∉ S
    • The empty set is a subset of every set, but NOT a member of every set: ∅∉{1}, ∅∈{1,∅}.
    • Set theories that do not allow sets to be members of themselves are called “well-founded.”
    • Set theories that do allow sets to be members of themselves are called “non-well-founded.”
    • Some set theories define membership between non-sets such as 5 ∈ 5 to mean 5 = 5, which would imply that 5 = {5} and {5} ∈ {5}. Other set theories would treat 5 ∈ 5 as undefined.

  5. Set Builder: C:{x|φ}
    • Set C contains all x where x a variable of formula φ and φ holds true for every x.
    • If φ is true when x is an even integer, then C:{x|φ} is the set of all even integers.

  6. Intersection (“and”): A ∩ B = {3}
    • The intersection of two sets is the set containing only elements that are members of both sets.
      • Two sets with an intersection that is the empty set are called disjoint sets or mutually exclusive.

  7. Union (“or”): A ∪ B = {1,2,3,4,5}
    • The union of two sets is the set containing elements that are members of either set.

  8. Compliment (“not”): Ac = {4,5,6}
    • The compliment of set A with regard to universe S is the set containing those elements not in A, but in S.
    • Compliments cancel like double negatives (e.g. not not red = red, Acc = A, Accc = Ac).

  9. Cardinality: S:{0,1,2,3,4,5}, |S| = 6
    • The cardinality of a set is the number of unique elements in the set.

De Morgan’s Law

De Morgan’s Law states that intersections are equivalent to unions (and vice versa) under compliments. It has two equivalent expressions (because either version implies the other – if unions are intersections, then intersections are unions).

  1. For sets A and B, the compliment of their union is the intersection of their compliments: (A ∪ B)c = Ac ∩ Bc.
  2. For sets A and B, the compliment of their intersection is the union of their compliments: (A ∩ B)c = Ac ∪ Bc.

Toggle the “and’s” and “or’s” and apply “not” to every individual element and to every operation to arrive at the equivalent statement to the one you started with. This applies to ANY intersection or compliment because any set theory formula can be expressed as a compliment: (A ∪ B) = (A ∪ B)cc.

You can view a proof of De Morgan’s law here.

Example of De Morgan’s Law

De Morgan’s law applies to any logical system using “and,” “or,” and “not” operations so it has many applications outside of math especially in computer programming. See an example below using a SQL query.

SQL
/* Both top and bottom subueries are equivalent according
   to De Morgan so full query will return 0 rows. */
select CustomerId, FirstName, LastName
from TblCustomer
where (
    not(FirstName = 'Frank')
    and LastName = 'Barnes'
)

except  /* 'except' removes results in bottom subquery 
           from results in top subquery */

select CustomerId, FirstName, LastName
from TblCustomer
where not (
    FirstName = 'Frank'
    or not(LastName = 'Barnes')
)

Space Is a Complex Set

Sets are only defined by membership; spaces are more complex. Space is a commonly used term with a few different definitions, but generally a space is a set with some additional structure built on or attached to it, which creates additional properties the set alone does not possess.

For example, by defining a metric function (a function giving the distance between two elements) on the set of all real numbers it becomes a metric space called the real number line. The elements in a metric space have a distance property (and therefore also order property); the members of the set of all real numbers do not posses order or distance properties.

Distance and Intervals

One of the most common structures that can be built on sets, especially those containing number systems, is a distance function. Distance can be used to establish order, which can then be used to define intervals. An interval is a subsets described as all the elements from a set between a lower and upper boundary.

An open boundary is represented by a parenthesis and indicates the boundary value is NOT contained in the set. A closed boundary is represented by a square bracket and indicates boundary value IS contained in the set. See some examples below.

\[\large{
\begin{array} {ll}
x:(1,6)= x:\{n|\;1<n<6\} && \normalsize\text{(open interval)} \\ \\
x:[1,6]= x:\{n|\;1\le n\le 6\} && \normalsize\text{(closed interval)} \\ \\
x:[1,6)= x:\{n|\;1\le n< 6\} && \normalsize\text{(mixed interval)} \\ \\
x:(-\infty,\infty)= x:\{n|\;-\infty<n<\infty\} && \normalsize\text{(unbounded interval)} \\ \\
[3,3) = (3,3] = [3,3] = (3,3) = \{3\}* && \normalsize\text{(degenerate interval*)}
\end{array}
}\]

*Generally, but not always, when you see a reference to an interval it should be assumed to be a proper, i.e. non-degenerate, interval.

Discrete Space Is Countable

Discrete space is space where the elements are separable from one another; every element can be isolated within a neighborhood around itself. There are multiple ways to construct this, but using distance is the most intuitive. A neighborhood around an element is defined using a lower and upper bound some distance away from the element itself.

In the example below 2 is isolated in the natural numbers within a neighborhood 0.5 distance from itself. In other words, the interval results in a singleton containing 2.

\[\large{
\begin{array}{c}
n\in\mathbb{N} \\ \\
x:[1.5,2.5] = x:\{n|\;1.5\le n\le 2.5\} =\{2\} \\ \\
\mathbb{N}:\Big\{0\;\;\;,\;\;\;1\;\;\;,\;\stackrel{\text{1.5}}{[}\;2\;\stackrel{\text{2.5}}{]}\;,\;\;\;3\;\;\;,\;\;\;4\;\;\;…\Big\}
\end{array}
}\]

Every element in discrete space can be isolated. This implies that any subset defined by bounded intervals has only a finite number of elements in it and that we can always identify explicitly the next closest element to a given element. In this way, even if a discrete set is infinite it is still countable.

Imagine a never ending staircase. Even though you will never reach the end by walking up them, the stairs are discrete (i.e. separate from one another) and you can count the number of stairs you have walked. You can also select any two stairs and count the stairs between them.

Discrete spaces are descriptions of countable collections of things (people, days, cars, etc.) even if those collections are themselves infinite.

Index Notation

Countable sets are be indexed by the natural numbers. When a set is countable its elements may be represented by a subscript variable:

\[\large{
X:\big\{X_1,X_2,X_3,…,X_n\big\}
}\]

Continuous Space Is Uncountable

glass of water

Continuous space is a set with a structure such that every element cannot be isolated from its neighbors using distance or any other method. The formal definitions here involve topology and other concepts that are beyond an elementary introduction. Informally, a continuous space is infinitely divisible space with the following properties:

  • Contains Points: Points are infinitely small. They can be thought of as addresses in continuous space rather than substantial objects.

    Being infinitely small, points can exist literally distance 0 from other points.
    \[
    \begin{array}{ll}
    r\in\mathbb{R} && \normalsize\text{(r is a real number.)} \\ \\
    \lim\limits_{d \to 0^{+}} (r+d) = r && \normalsize\text{(d approaches 0 from the right, but is never 0.)} \\ \\
    r \;- \Big(\lim\limits_{d \to 0^{+}} (r+d)\Big) = 0 && \normalsize\text{(The next number after r is distance 0 from r.)}
    \end{array}
    \]
  • Is Dense: No points can be isolated in continuous space. No matter how close you make the bounds of an interval around a point, there are always other points in the interval.
    \[
    \begin{array}{l}
    [1.5,2.5] = \{1.5,…,2.0,…,2.5\} \\ \\
    [1.9,2.1] = \{1.9,…,2.0,…,2.1\} \\ \\
    [1.9999,2.0001] = \{1.99,…,2.0,…,2.01\}
    \end{array}
    \]
  • Has Total order: Every two points have an order, even if they are infinitely close together (i.e. even if the distance between the points is 0).
  • Is a Continuum: There are no “holes” or “gaps” in continuous space. If you define a set using a closed interval and either the upper or lower bound is missing from the set, then that value is missing from the space (i.e. the space has a “hole”).

Non-Countable Infinity of Infinities

A useful exercise for contrasting countable space with continuous space is to try and count continuous space (keeping the properties of continuous space in mind).

Continuous space has total order, therefore we know there is a number immediately “after” 2.0, but we also know that that number is literally 0 distance from 2.0 (otherwise we could isolate 2 according to distance). And so counting would look something like this: 2.0, the number after 2.0, the number after the number after 2.0, etc. There is, a countable infinity AT each point, and each point in that infinity is 0 distance from the starting point. Counting such an infinity will not move you any distance away from your starting value.

From the exercise above and the properties of continuous space it should be clear that any interval in continuous space will contain a non-countable infinity of points even if the total distance of the interval itself is finite (e.g. there is a non-countable infinity of points between 0.0 and 1.0 on the real number line). For this reason, continuous space is also called infinitely divisible space and represents things such as distance, area, time, frequency, angles, etc.

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