Set Theory Symbols

A comprehensive collection of the most notable symbols in set theory, categorized by function into tables along with each symbol's meaning and example.

Set theory is a branch of mathematics dedicated to the study of collections of objects, its properties, and the relationship between them. The following list documents some of the most notable symbols in set theory, along each symbol’s usage and meaning.

For readability purpose, these symbols are categorized by their function into tables. Other comprehensive lists of symbols — as categorized by subject and type — can be also found in the relevant pages below (or in the navigational panel).

Cover of Math Vault's Comprehensive List of Mathematical Symbols eBook

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Get the master summary of mathematical symbols in eBook form — along with each symbol’s usage and LaTeX code.

Constants

In set theory, constants are often one-character symbols used to denote key mathematical sets. The following table documents the most notable of these — along with their respective meaning and example.

Symbol NameExplanationExample
$\varnothing$, $\emptyset$, $\{ \}$Empty set$|\varnothing|=0$
$\mathbb{N}$Set of natural numbers$1 \in \mathbb{N}$
$\mathbb{Z}$Set of integersIf $m, n \in \mathbb{Z}$, then $m+n, mn \in \mathbb{Z}$.
$\mathbb{Q}$Set of rational numbers$\pi \notin \mathbb{Q}$
$\mathbb{R}$Set of real numbers$e \in \mathbb{R}$
$\mathbb{C}$Set of complex numbers$\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}$
$\mathbb{U}, U$Universal setWhen $U = \mathbb{R}$, $\overline{\mathbb{Q}} = \mathbb{R} \setminus \mathbb{Q}$.
Hierarchy of Key Sets of Numbers NZQRC
Hierarchy of Key Sets of Numbers

For other key sets of numbers, see key mathematical sets in algebra.

Variables

Similar to other fields in mathematics, set theory often uses a designated list of variable symbols to refer to varying objects and quantities. The following table documents the most common of these — along with their respective example and meaning.

Symbol NameUsed ForExample
$A ,B, C$Sets$A \subseteq B \cup C$
$a, b, c$Elements of setIf $a \in A$ and $b \in B$, then $a, b \in A \cup B$.
$\alpha, \beta, \gamma$Ordinal numbersIf $P(\beta)$ for all $\beta < \alpha$ implies $P(\alpha)$, for all $\alpha$, then $P$ holds in general by transfinite induction.
$\lambda$Limit ordinals$\lambda$ is a limit ordinal if it’s neither $0$ nor a successor ordinal.
$\kappa$Cardinal numbersFor each finite cardinal $\kappa$, its successor is simply $\kappa + 1$.

Delimiters

In set theory, delimiters are symbols used to delineate the separation between independent mathematical entities, and often occur in the context of definition of sets. The following table documents the most common of these — along with their respective usage and meaning.

Symbol NameExplanationExample
$\{ \}$Indicator for sets$\{5, e, \pi, 1\}$
$( )$Indicator for tuples$(1, 2.89) \in \mathbb{N} \times \mathbb{R}$
$|, :$‘Such that’ marker in set-builder notation$\{ x^2 : x \in \mathbb{Z} \} =$
$\{ y \in \mathbb{Z} \mid y = x^2$
$\mathrm{for \ some \ } x \in \mathbb{Z} \} $

Operators

Since the concept of set lies at the foundation of mathematics, many set operators — which allow one to refer to new sets through pre-existing sets — exist. The following table documents the most notable of these — along with their respective example and meaning.

Symbol NameExplanationExample
$\overline{A}, A^{\mathrm{c}}, A’$Complement of set $A$$\overline{A} = \{ x \in U \mid \\ x \notin A \}$
$A \cap B$Intersection of sets $A$ and $B$$\mathbb{R} \cap \mathbb{I} = \{ 0 \}$
$\displaystyle \bigcap_{i=m}^n A_i$, $\, \displaystyle \bigcap_{i \in I} A_i$Generalized intersection of sets $A_i$$\displaystyle \bigcap_{i=1}^{\infty} \, ( 0, 1 / i ) = \varnothing$
$A \cup B$Union of sets $A$ and $B$$A \cup (B \cup C) =$
$(A \cup B) \cup C$
$\displaystyle \bigcup_{i=m}^n A_i$, $\, \displaystyle \bigcup_{i \in I} A_i$Generalized union of sets $A_i$$\displaystyle \bigcup_{i=\{3, 5 , 8\}} [-i, i] = [-8, 8]$
$A \sqcup B$Disjoint union of sets $A$ and $B$If $A = \{ 2, 5 \}$ and $B = \{ 1 ,2 \}$, then $A \sqcup B = \{ (2, a), (5, a),$ $(1,b), (2,b) \}.$
$\displaystyle \bigsqcup_{i=m}^n A_i$, $\, \displaystyle \bigsqcup_{i \in I} A_i$Generalized disjoint union of sets $A_i$$\displaystyle \bigsqcup_{i=1}^n A_i = \bigcup_{i=1}^n$
$\{ (a_i, i) \mid a_i \in A_i \}$
$A \setminus B$, $A-B$Set difference
($A$ minus $B$)
$\{ 5, 6 ,8\} \setminus \{6, 9\} =$
$\{5, 8\}$
$A \, \triangle \, B$, $A \ominus B$Symmetric difference of sets $A$ and $B$$A \, \triangle \, B =$
$(A \setminus B) \cup (B \setminus A)$
$A \times B$Cartesian product
($A$ times $B$)
In general, $A \times B \ne B \times A$.
$A^n$$n$th Cartesian power of set $A$$\mathbb{R}^3 = \mathbb{R} \times \mathbb{R} \times \mathbb{R}$
$\displaystyle \prod_{i=m}^n A_i$Generalized Cartesian product of sets $A_i$$\displaystyle \prod_{i=1}^n \{i\} = \{ (1, \ldots, n) \}$
$\mathcal{P}(A)$Power set of $A$$\mathcal{P}(\{a, b\}) =$
$\left\{ \varnothing, \{a\}, \{b\}, \{a, b\} \right\}$
$2^{A}$Set of functions from $A$ to a two-element setFor all sets $A$, $|2^A| = |\mathcal{P}(A)$|.
$B^A$Set of functions from $A$ to $B$For all finite sets $A$ and $B$, $|B^A| = |B|^{|A|}$.

The following diagrams, known as Venn diagrams, provide a visualization of five of the key sets mentioned above (where $A$ and $B$ are represented by the left and the right circle, respectively).

Relational Symbols

In set theory, relational symbols are often used to describe relationships between sets, or relationships between a set and its element. The following table documents the most notable of these symbols — along with their respective meaning and example.

Symbol NameExplanationExample
$x \in A$Set membership
(Element $x$ is in set $A$)
$i \in \mathbb{C}$
$x \notin A$Set non-membership
(Element $x$ is not in set $A$)
$\pi \notin \mathbb{Q}$
$A \ni x$Set inclusion
(Set $A$ includes element $x$)
$A \ni x \iff x \in A$
$A \not\ni x$Set non-inclusion
(Set $A$ does not include element $x$)
$\{a\} \not\ni \{a\}$
$A = B$Set equivalence
(Sets $A$ and $B$ are equal)
$\varnothing = \{ \}$
$A \subseteq B$Subset relation
($A$ is a subset of $B$)
$\mathbb{N} \subseteq \mathbb{Q}$
$A \subset B$Proper subset relation
($A$ is a proper subset of $B$)
$A \subset B \iff$
$A \subseteq B \ \mathrm{and}\ A \ne B$
$A \not\subseteq B$Non-subset relation
($A$ is not a subset of $B$)
If $A \not\subseteq B$, then there exists an $x \in A$ such that $x \notin B$.
$A \supseteq B$Superset relation
($A$ is a superset of $B$)
$\{5, 8, 1\} \supseteq \{1, 8\}$
$A \supset B$Proper superset relation
($A$ is a proper superset of $B$)
$A \supset B \iff \\ B \subset A$
$A \not\supseteq B$Non-superset relation
($A$ is not a superset of $B$)
$\mathbb{N} \not\supseteq \mathbb{R}$

Cardinality-related Symbols

In set theory, the concept of cardinality provides a way of quantifying and comparing the sizes of different sets. The following table documents some of the most notable symbols related to cardinality — along with each symbol’s usage and meaning.

(For a refresher on Greek and Hebrew symbols, see Greek, Hebrew and Latin-based symbols.)

Symbol NameExplanationExample
$|A|$Cardinality of set $A$$|\mathbb{N}|=|\mathbb{Z}|$
$\aleph_0$, $\aleph_1, \ldots$Aleph numbersUnder continuum hypothesis, $\aleph_1 = 2^{\aleph_0}$.
$\beth_0, \beth_1, \ldots$Beth numbers$\beth_0 = |\mathbb{N}|,$
$\beth_1=|\mathcal{P}(\mathbb{N})|,$
$\beth_2 = |\mathcal{P}(\mathcal{P}(\mathbb{N}))|$
$\gimel$Gimel function$\gimel(\kappa) > \kappa$
$\mathfrak{c}$Cardinality of the continuum$|\mathbb{R}| = \mathfrak{c} = |\mathbb{\mathcal{P}(\mathbb{N})}|$
$\omega_0, \omega_1, \ldots$Omega ordinals$\omega = \{ 0, 1, 2, \ldots \},$
$\omega + 1 = \omega \cup \{ \omega \}$
$\Omega$First uncountable ordinal$\Omega = \omega_1$
$\varepsilon_0, \varepsilon_1, \ldots$Epsilon ordinals$\varepsilon_0 = \omega^{\omega^{\omega^{\ \cdot^{\,\cdot^{\,\cdot}}}}}$
$S(\alpha)$Successor of ordinal $\alpha$$\alpha + S(\beta) = \\ S (\alpha + \beta)$
$\kappa^+$Successor of cardinal $\kappa$$\aleph_{\alpha}^+ = \aleph_{\alpha+1}$

For the master list of symbols, see mathematical symbols. For lists of symbols categorized by type and subject, refer to the relevant pages below for more.

Cover of Math Vault's Comprehensive List of Mathematical Symbols eBook

Prefer the PDF version instead?

Get the master summary of mathematical symbols in eBook form — along with each symbol’s usage and LaTeX code.

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