Set Theory: Basic Concepts, Symbols and Examples (2025)

In this section we will develop the most important concepts of discrete mathematics set theory that serve as a basis for various areas of study in mathematics.

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What is set theory?

Set theory is a branch of mathematics that studies the properties and relationships of sets, which are well-defined collections of objects. These sets can contain numbers, letters, functions, or other sets, depending on the context.

Set theory was developed in the late 19th century by mathematicians such as Georg Cantor and Richard Dedekind and provides a formal framework for constructing other mathematical structures. It is widely used in virtually all fields of mathematics, as well as in logic and computer science.

Basic concepts

The fundamental concepts of set theory are those of set, element, and membership. Based on these, cardinality, inclusion relations, set equality, and other operations are defined.

Set

A set is a collection of well-defined and distinct objects. These objects can be concrete (numbers, letters) or abstract (ideas, concepts).

Sets are denoted by uppercase letters A, B, C, etc. The elements are written inside braces {}, separated by commas.

Examples:

• A = {1, 2, 3}
• B = {a, b, c}
• C = {red, green, blue}

Element

An element is an object that is part of a set. It can be anything, such as a number, a letter, a person, an animal, a physical object, or even an abstract concept.

Two equal elements in the same set are considered as one. Additionally, it does not matter in which order the elements are written; the set remains the same.

Examples:

• Set A = {1, 2, 3} has three elements: the number 1, the number 2, and the number 3.
• Set B = {a, b, c} has three elements: the letter "a", the letter "b", and the letter "c".
• Set C = {red, green, blue} has three elements: the color red, the color green, and the color blue.

Belonging

Belonging or membership is the relationship an element has with a set, whether it belongs to it or not. To indicate that an element belongs to the set, the symbol ∈ is used; otherwise, the symbol ∉ is used.

Examples:

• 1 ∈ A (the number 1 belongs to set A)
• c ∈ B (the letter "c" belongs to set B)
• red ∈ C (the color red belongs to set C)
• 4 ∉ A (the number 4 does not belong to set A)
• d ∉ B (the letter "d" does not belong to set B)

Cardinality

The cardinality of a set is a measure that indicates how many elements that set has. In other words, it is the number of distinct elements contained in the set. The cardinality of a set A is symbolized by Card(A) or |A|.

Examples:

• The cardinality of set B = {a, b, c} is |B| = 3, because it has three elements.
• The cardinality of set D = {1, 2, e, f, g} is |D| = 5, because it has five elements.

The cardinality of a set can be finite or infinite. In finite sets, like the examples above, the cardinality is simply the number of elements the set contains. In infinite sets, such as the set of all natural numbers (1, 2, 3, 4, ...), the cardinality is infinite and is expressed using special symbols, such as ℵ₀ (aleph-zero).

More information and examples of set cardinality

Set equality

Two sets are equal if they have exactly the same elements, regardless of the order in which they are written.

Examples:

• A = {1, 2, 3} and E = {3, 1, 2} are equal sets.
• C = {red, green, blue} and F = {red, blue, green} are equal sets.
• B = {a, b, c} and G = {a, b, c, d} are not equal sets because they do not have the same elements.

More examples of set equality

Set inclusion

If at least one element of A is not within B, it is said that A is not a subset of B, and it is written as A⊈B. If it happens that every element of A is in B, but B has some element that is not in A, it is said that A is a proper subset of B and is symbolized as A⊂B.

Examples:

• A = {1, 2} is a subset of B = {1, 2, 3, 4, 5}, because every element of A is also an element of B: A⊆B. It is also true that A⊂B, because B has elements that are not in A.
• C = {red, green} is a subset of D = {red, green, blue}, meaning C ⊆ D.

More information and examples about set inclusion

Set operations

There are several operations that can be performed between two or more sets, also called set algebra, they are:

• Union:the union of two sets A and B is the set that contains all elements that are in A or in B or in both. It is denoted by A ∪ B.
• Intersection:the intersection of two sets A and B is the set that contains all elements that are in both A and B. It is denoted by A ∩ B.
• Difference:the difference of two sets A and B is the set that contains all elements that are in A but not in B. It is denoted by A-B.
• Symmetric difference: the symmetric difference between two sets A and B is the set formed by the elements of A that do not belong to B and the elements of B that do not belong to A, that is, those elements that belong exclusively to one of the sets. It is denoted by A Δ B.
• Complementation: the complement of a set A, denoted as A', consists of all elements of the universal set U that are not in A. The universal set is the one that contains all the relevant elements for the particular context of study.

More information and examples about set operations

Types of sets

In set theory, there are several types of sets, some of which are:

• Finite set:a set that has a finite number of elements.
• Infinite set:a set that has an infinite number of elements.
• Empty set:the set that has no elements. It is denoted by Ø.
• Singleton set: a set that contains exactly one element.
• Universal set: a set that contains all the relevant elements of a given context. It is typically denoted as U.
• Power set: the power set of A, denoted as P(A), is the set that contains all possible subsets of A, including the empty set and A itself.

Some special sets that can be highlighted are the numeric sets, which contain numbers with certain common properties: natural numbers (N), integers (Z), rationals (Q), irrationals (I), real numbers (R), complex numbers (C).

More types of sets with examples

Venn diagrams

Sets are often represented graphically by so-called Venn diagrams. These diagrams are closed flat figures; typically, the universal set is represented by the interior of a rectangle and the other sets by bubbles inside the rectangle. The corresponding elements are located within these figures.

For example, for the sets A = {a, b, c} and B = {e, f}, the Venn diagram is as follows:

Notation

The following tables will display all the necessary symbols to work in set theory along with their meanings.

Basic set symbols and operations

Predicate logic symbols for sets

Numeric sets