Set Theory

Set theory is one of the fundamental branches of mathematics. It provides the language used to describe collections of objects and forms the foundation of many mathematical disciplines, from algebra and analysis to probability and topology.

At the heart of set theory lies the concept of a set, a simple yet powerful idea that allows mathematicians to organize, classify, and study mathematical objects in a rigorous way.

What Is a Set?

A set is a collection of objects, called elements or members, that share a common property. A set may also be described as a class, collection, aggregate, or family.

The notion of a set is considered a primitive concept in mathematics because it is not usually defined in terms of simpler ideas.

One important characteristic of a set is that the order of its elements is irrelevant. A set remains the same regardless of how its elements are arranged.

What Makes a Collection a Mathematical Set?

In mathematics, a set is a collection of objects defined by an objective membership criterion, a rule that makes it possible to determine unambiguously whether an object belongs to the collection.

This requirement distinguishes mathematical sets from ordinary collections based on personal preferences or opinions.

Whenever the term "set" is used in this discussion, it refers to a mathematical set.

Example. The set of prime numbers is a mathematical set because membership is determined by a precise rule. It contains all natural numbers greater than 1 that are divisible only by 1 and themselves. $$ P = \{ 2, 3, 5, 7, 11, ... \} $$ By contrast, the collection of the best songs is not a mathematical set because different people may disagree about which songs should belong to it.

How Sets Are Represented

Mathematicians use several different ways to represent sets, depending on the context and the amount of information they want to convey.

Sets are usually denoted by uppercase letters such as A, B, C, and D.

Their elements are generally denoted by lowercase letters such as a, b, c, and d.

Membership and Non-membership Symbols

The symbol ∈ indicates that an element belongs to a set.

It is read as "x belongs to A" or "x is an element of A".

The symbol ∉ indicates that an element does not belong to a set.

It is read as "x does not belong to A" or "x is not an element of A".

Example. Consider the set B of even numbers between 1 and 9. $$ B = \{ 2, 4, 6, 8 \} $$ Since 2 is an even number, it belongs to B: $$ 2 \in B $$ Since 3 is odd, it does not belong to B: $$ 3 \notin B $$

Graphical Representation

Sets can be represented visually using Euler diagrams, Venn diagrams, or, in elementary contexts, Euler-Venn diagrams.

Each set is represented by a closed region, usually a circle or an oval.

Elements belonging to the set are placed inside the region, while elements outside the region do not belong to the set.

Note. In the diagram above, set A contains the natural numbers up to 10, while set B contains the prime numbers between 11 and 20. Numbers outside both regions belong to neither set.

If an element belongs to two sets simultaneously, it is placed in the overlapping region shared by both sets.

For example, if set B represented the prime numbers between 1 and 20, then sets A and B would intersect.

In this case, the elements common to both sets, A⋂B={1, 2, 3, 5, 7}, appear in the intersection.

Note. These diagrams are associated with the mathematicians Leonhard Euler and John Venn, whose work helped establish visual methods for representing sets and their relationships.

Roster Notation

One of the simplest ways to represent a set is to list all of its elements explicitly.

The elements are written inside curly braces and separated by commas.

This representation is known as roster notation or enumerative notation.

Because the order of elements is irrelevant, the same set can be written in many different ways.

Likewise, repeated elements are ignored because each element can appear only once in a set.

Roster notation works particularly well for small finite sets but quickly becomes impractical when the number of elements grows.

Note. Ellipses (...) are often used when the pattern is obvious. For example, A={1,2,...,10} denotes the natural numbers from 1 to 10. $$ A = \{ 1,2, ... ,10 \} $$ Likewise, A={1,2,3,...} denotes the set of all natural numbers. $$ A = \{ 1,2,3, ... \} $$

Set-builder Notation

When a set contains many elements, or infinitely many elements, it is often more convenient to describe it through a defining property.

This approach is known as set-builder notation.

Instead of listing every element, this notation specifies the condition that elements must satisfy to belong to the set.

Example. The natural numbers from 1 to 10 can be represented as A={x∈N:0<x≤10}.

Here, x is a variable ranging over the natural numbers. The colon (:) is read as "such that", although the vertical bar | is often used instead. $$ A = \{ x \in \mathbb{N} \mid 0 < x \le 10 \} $$ This expression can be read as: "A is the set of all natural numbers x such that 0 < x ≤ 10."

Choosing the Right Representation

Set-builder notation can describe both finite and infinite sets, provided that their elements can be characterized by one or more defining properties.

Roster notation is most effective when the elements can be listed conveniently.

Graphical representations are particularly useful when the goal is to visualize relationships such as inclusion, intersection, or overlap between sets.

Finite and Infinite Sets

Sets can be classified according to the number of elements they contain.

• A finite set contains a finite number of elements.

  Example. The set of books in a backpack is finite. The set of grains of sand in a container is also finite, even though it may contain a very large number of elements.

• An infinite set contains infinitely many elements.

  Example. The set of natural numbers is infinite because every natural number n has a successor, n+1. Consequently, there is no greatest natural number.

Sets of Numbers

Many of the most important sets in mathematics are numerical sets.

By convention, specific uppercase letters are used to denote them.

• N denotes the set of natural numbers
• Z denotes the set of integers
• Q denotes the set of rational numbers
• I denotes the set of irrational numbers
• R denotes the set of real numbers
• C denotes the set of complex numbers

The Empty Set

The empty set is the unique set that contains no elements. It is denoted by the symbol ∅. $$ \emptyset = \{ \} $$

The empty set may also be represented by a pair of empty curly braces.

Although it contains no elements, it plays an important role in mathematics because it is a subset of every set.

There is only one empty set. For example, the set of odd natural numbers divisible by two is empty because no odd number is divisible by two. Likewise, the set of squares with three sides is empty because squares have four sides. Although these collections are described differently, they both correspond to the same empty set.

And so on.