Glossary of mathematical symbols

A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other sort of mathematical objects. As the number of these sorts has dramatically increased in modern mathematics, the Greek alphabet and some Hebrew letters are also used. In mathematical formulas, the standard typeface is italic type for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. For having more symbols, other typefaces are also used, mainly boldface $\mathbf {a,A,b,B} ,\ldots$ , script typeface ${\mathcal {A,B}},\ldots$ (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur ${\mathfrak {a,A,b,B}},\ldots$ , and blackboard bold $\mathbb {N,Z,Q,R,C,H,F} _{q}$ (the other letters are rarely used in this face, or their use is unconventional).

The use of Latin and Greek letters as symbols for denoting mathematical objects is not described in this article. For such uses, see Variable (mathematics) and List of mathematical constants. However, some symbols that are described here have the same shape as the letter from which they are derived, such as $\textstyle \prod {}$ and $\textstyle \sum {}$ .

Letters are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin in punctuation marks and diacritics traditionally used in typography. Other, such as $+$ and $=$ , have been specially designed for mathematics, often by deforming some letters, as in the cases of $\in$ and $\forall$ .

Layout[]

Normally, entries of a glossary are structured by topics and sorted alphabetically. This is not possible here, as there is no natural order on symbols, and many symbols are used in different parts of mathematics with different meanings, often completely unrelated. Therefore some arbitrary choices had to be made, which are summarized below.

The article is split into sections that are sorted by an increasing level of technicality. That is, the first sections contain the symbols that are encountered in most mathematical texts, and that are supposed to be known even by beginners. On the other hand, the last sections contain symbols that are specific to some area of mathematics and are ignored outside these areas. However, the long section on brackets has been placed near to the end, although most of its entries are elementary: this makes it easier to search for a symbol entry by scrolling.

Most symbols have multiple meanings that are generally distinguished either by the area of mathematics where they are used or by their syntax, that is, by their position inside a formula and the nature of the other parts of the formula that are close to them.

As readers may not be aware of the area of mathematics to which is related the symbol that they are looking for, the different meanings of a symbol are grouped in the section corresponding to their most common meaning.

When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. For summarizing the syntax in the entry name, the symbol $\Box$ is used for representing the neighboring parts of a formula that contains the symbol. See § Brackets for examples of use.

Most symbols have two printed versions. They can be displayed as Unicode characters, or in LaTeX format. With the Unicode version, using search engines and copy-pasting are easier. On the other hand, the LaTeX rendering is often much better (more aesthetic), and is generally considered a standard in mathematics. Therefore, in this article, the Unicode version of the symbols is used (when possible) for labelling their entry, and the LaTeX version is used in their description. So, for finding how to type a symbol in LaTeX, it suffices to look at the source of the article.

For most symbols, the entry name is the corresponding Unicode symbol. So, for searching the entry of a symbol, it suffices to type or copy the Unicode symbol into the search textbox. Similarly, when possible, the entry name of a symbol is also an anchor, which allows linking easily from another Wikipedia article. When an entry name contains special characters such as [, ], and |, there is also an anchor, but one has to look at the article source to know it.

Finally, when there is an article on the symbol itself (not its mathematical meaning), it is linked to in the entry name.

Arithmetic operators[]

$+$: 1. Denotes addition and is read as plus; for example, $3 + 2$ .; 2. Sometimes used instead of $\sqcup$ for a disjoint union of sets.
$-$: 1. Denotes subtraction and is read as minus; for example, $3 - 2$ .; 2. Denotes the additive inverse and is read as negative or the opposite of; for example, $-2$ .; 3. Also used in place of $\$ for denoting the set-theoretic complement; see \ in § Set theory.
$\times$: 1. In elementary arithmetic, denotes multiplication, and is read as times; for example, $3 \times 2$ .; 2. In geometry and linear algebra, denotes the cross product.; 3. In set theory and category theory, denotes the Cartesian product and the direct product. See also × in § Set theory.
$\cdot$: 1. Denotes multiplication and is read as times; for example, $3 \cdot 2$ .; 2. In geometry and linear algebra, denotes the dot product.; 3. Placeholder used for replacing an indeterminate element. For example, "the absolute value is denoted $| \cdot |$ " is clearer than saying that it is denoted as $| |$ .
$\pm$: 1. Denotes either a plus sign or a minus sign.; 2. Denotes the range of values that a measured quantity may have; for example, $10 \pm 2$ denotes a unknown value that lies between 8 and 12.
$\mp$: Used paired with $\pm$ , denotes the opposite sign; that is, $+$ if $\pm$ is $-$ , and $-$ if $\pm$ is $+$ .
$\div$: Widely used for denoting division in anglophone countries, it is no longer in common use in mathematics and its use is "not recommended".^[1] In some countries, it can indicate subtraction.
$:$: 1. Denotes the ratio of two quantities.; 2. In some countries, may denote division.; 3. In set-builder notation, it is used as a separator meaning "such that"; see ${□ : □}$ .
$/$: 1. Denotes division and is read as divided by or over. Often replaced by a horizontal bar. For example, $3 / 2$ or ${\frac {3}{2}}$ .; 2. Denotes a quotient structure. For example, quotient set, quotient group, quotient category, etc.; 3. In number theory and field theory, $F/E$ denotes a field extension, where $F$ is an extension field of the field $E$ .; 4. In probability theory, denotes a conditional probability. For example, $P(A/B)$ denotes the probability of $A$ , given that $B$ occurs. Also denoted $P(A\mid B)$ : see " $|$ ".
$\sqrt$: Denotes square root and is read as the square root of. Rarely used in modern mathematics without an horizontal bar delimiting the width of its argument (see the next item). For example, $\sqrt2$ .
$\sqrt$: 1. Denotes square root and is read as the square root of. For example, ${\sqrt {3+2}}$ .; 2. With an integer greater than 2 as a left superscript, denotes an $n$ th root. For example, ${\sqrt[{7}]{3}}$ .
$^$: 1. Exponentiation is normally denoted with a superscript. However, $x^{y}$ is often denoted $x^y$ when superscripts are not easily available, such as in programming languages (including LaTeX) or plain text emails.; 2. Not to be confused with ∧.

Equality, equivalence and similarity[]

$=$: 1. Denotes equality.; 2. Used for naming a mathematical object in a sentence like "let $x=E$ ", where $E$ is an expression. On a blackboard and in some mathematical texts, this may be abbreviated as $x\,{\stackrel {\mathrm {def} }{=}}\,E$ . This is related to the concept of assignment in computer science, which is variously denoted (depending on the used programming language) $=,:=,==,\leftarrow ,\ldots$
$\neq$: Denotes inequality and means "not equal".
$\approx$: Means "is approximately equal to". For example, $\pi \approx {\frac {22}{7}}$ (for a more accurate approximation, see pi).
$~$: 1. Between two numbers, either it is used instead of $\approx$ to mean "approximatively equal", or it means "has the same order of magnitude as".; 2. Denotes the asymptotic equivalence of two functions or sequences.; 3. Often used for denoting other types of similarity, for example, matrix similarity or similarity of geometric shapes.; 4. Standard notation for an equivalence relation.; 5. In probability and statistics, may specify the probability distribution of a random variable. For example, $X\sim N(0,1)$ means that the distribution of the random variable $X$ is standard normal.^[2]
$\equiv$: 1. Denotes an identity, that is, an equality that is true whichever values are given to the variables occurring in it.; 2. In number theory, and more specifically in modular arithmetic, denotes the congruence modulo an integer.
$\cong$: 1. May denote an isomorphism between two mathematical structures, and is read as "is isomorphic to".; 2. In geometry, may denote the congruence of two geometric shapes (that is the equality up to a displacement), and is read "is congruent to".

Comparison[]

$<$: 1. Strict inequality between two numbers; means and is read as "less than".; 2. Commonly used for denoting any strict order.; 3. Between two groups, may mean that the first one is a proper subgroup of the second one.
$>$: 1. Strict inequality between two numbers; means and is read as "greater than".; 2. Commonly used for denoting any strict order.; 3. Between two groups, may mean that the second one is a proper subgroup of the first one.
$\leq$: 1. Means "less than or equal to". That is, whatever $A$ and $B$ are, $A \leq B$ is equivalent to $A < B or A = B$ .; 2. Between two groups, may mean that the first one is a subgroup of the second one.
$\geq$: 1. Means "greater than or equal to". That is, whatever $A$ and $B$ are, $A \geq B$ is equivalent to $A > B or A = B$ .; 2. Between two groups, may mean that the second one is a subgroup of the first one.
$≪ , ≫$: 1. Mean "much less than" and "much greater than". Generally, much is not formally defined, but means that the lesser quantity can be neglected with respect to the other. This is generally the case when the lesser quantity is smaller than the other by one or several orders of magnitude.; 2. In measure theory, $\mu \ll \nu$ means that the measure $\mu$ is absolutely continuous with respect to the measure $\nu$ .
$≦$: 1. A rarely used synonym of $\leq$ . Despite the easy confusion with $\leq$ , some authors use it with a different meaning.
$≺ , ≻$: Often used for denoting an order or, more generally, a preorder, when it would be confusing or not convenient to use $<$ and $>$ .

Set theory[]

$\emptyset$: Denotes the empty set, and is more often written $\emptyset$ . Using set-builder notation, it may also be denoted ${ }.$
$#$: 1. Number of elements: $\#S$ may denote the cardinality of the set $S$ . An alternative notation is $|S|$ ; see $| □ |$ .; 2. Primorial: $n\#$ denotes the product of the prime numbers that are not greater than $n$ .; 3. In topology, $M\#N$ denotes the connected sum of two manifolds or two knots.
$\in$: Denotes set membership, and is read "in" or "belongs to". That is, $x\in S$ means that $x$ is an element of the set $S$ .
$\notin$: Means "not in". That is, $x\notin S$ means $\neg (x\in S)$ .
$\subset$: Denotes set inclusion. However two slightly different definitions are common. It seems that the first one is more commonly used in recent texts, since it allows often avoiding case distinctions.; 1. $A\subset B$ may mean that $A$ is a subset of $B$ , and is possibly equal to $B$ ; that is, every element of $A$ belongs to $B$ ; in formula, $\forall x,\,x\in A\Rightarrow x\in B$ .; 2. $A\subset B$ may mean that $A$ is a proper subset of $B$ , that is the two sets are different, and every element of $A$ belongs to $B$ ; in formula, $A\neq B\land \forall x,\,x\in A\Rightarrow x\in B$ .
$\subseteq$: $A\subseteq B$ means that $A$ is a subset of $B$ . Used for emphasizing that equality is possible, or when the second definition is used for $A\subset B$ .
$⊊$: $A\subsetneq B$ means that $A$ is a proper subset of $B$ . Used for emphasizing that $A\neq B$ , or when the first definition is used for $A\subset B$ .
$\supset , \supseteq , ⊋$: The same as the preceding ones with the operands reversed. For example, $B\supset A$ is equivalent to $A\subset B$ .
$\cup$: Denotes set-theoretic union, that is, $A\cup B$ is the set formed by the elements of $A$ and $B$ together. That is, $A\cup B=\{x\mid (x\in A)\lor (x\in B)\}$ .
$\cap$: Denotes set-theoretic intersection, that is, $A\cap B$ is the set formed by the elements of both $A$ and $B$ . That is, $A\cap B=\{x\mid (x\in A)\land (x\in B)\}$ .
$\$: Set difference; that is, $A\setminus B$ is the set formed by the elements of $A$ that are not in $B$ . Sometimes, $A-B$ is used instead; see – in § Arithmetic operators.
$⊖$: Symmetric difference: that is, $A\ominus B$ is the set formed by the elements that belong to exactly one of the two sets $A$ and $B$ . Notation $A\operatorname {\Delta } B$ is also used; see Δ.
$∁$: 1. With a subscript, denotes a set complement: that is, if $B\subseteq A$ , then $\complement _{A}B=A\setminus B$ .; 2. Without a subscript, denotes the absolute complement; that is, $\complement A=\complement _{U}A$ , where $U$ is a set implicitly defined by the context, which contains all sets under consideration. This set $U$ is sometimes called the universe of discourse.
$\times$: See also × in § Arithmetic operators.; 1. Denotes the Cartesian product of two sets. That is, $A\times B$ is the set formed by all pairs of an element of $A$ and an element of $B$ .; 2. Denotes the direct product of two mathematical structures of the same type, which is the Cartesian product of the underlying sets, equipped with a structure of the same type. For example, direct product of rings, direct product of topological spaces.; 3. In category theory, denotes the direct product (often called simply product) of two objects, which is a generalization of the preceding concepts of product.
$⊔$: Denotes the disjoint union. That is, if $A$ and $B$ are two sets, $A\sqcup B=A\cup C$ , where $C$ is a set formed by the elements of $B$ renamed to not belong to $A$ .
${\textstyle \coprod }$: 1. An alternative to $\sqcup$ for denoting disjoint union.; 2. Denotes the coproduct of mathematical structures or of objects in a category.

Basic logic[]

Several logical symbols are widely used in all mathematics, and are listed here. For symbols that are used only in mathematical logic, or are rarely used, see List of logic symbols.

$\neg$: Denotes logical negation, and is read as "not". If $E$ is a logical predicate, $\neg E$ is the predicate that evaluates to true if and only if $E$ evaluates to false. For clarity, it is often replaced by the word "not". In programming languages and some mathematical texts, it is sometimes replaced by " $~$ " or " $!$ ", which are easier to type on some keyboards.
$\lor$: 1. Denotes the logical or, and is read as "or". If $E$ and $F$ are logical predicates, $E\lor F$ is true if either $E$ , $F$ , or both are true. It is often replaced by the word "or".; 2. In lattice theory, denotes the join or least upper bound operation.; 3. In topology, denotes the wedge sum of two pointed spaces.
$\land$: 1. Denotes the logical and, and is read as "and". If $E$ and $F$ are logical predicates, $E\land F$ is true if $E$ and $F$ are both true. It is often replaced by the word "and" or the symbol " $&$ ".; 2. In lattice theory, denotes the meet or greatest lower bound operation.; 3. In multilinear algebra, geometry, and multivariable calculus, denotes the wedge product or the exterior product.
$⊻$: Exclusive or: if $E$ and $F$ are two Boolean variables or predicates, $E\veebar F$ denotes the exclusive or. Notations $E XOR F$ and $E\oplus F$ are also commonly used; see ⊕.
$\forall$: 1. Denotes universal quantification and is read "for all". If $E$ is a logical predicate, $\forall xE$ means that $E$ is true for all possible values of the variable $x$ .; 2. Often used improperly^[3] in plain text as an abbreviation of "for all" or "for every".
$\exists$: 1. Denotes existential quantification and is read "there exists ... such that". If $E$ is a logical predicate, $\exists xE$ means that there exists at least one value of $x$ for which $E$ is true.; 2. Often used improperly^[3] in plain text as an abbreviation of "there exists".
$\exists!$: Denotes uniqueness quantification, that is, $\exists !xP$ means "there exists exactly one $x$ such that $P$ (is true)". In other words, $\exists !xP(x)$ is an abbreviation of $\exists x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x))$ .
$\Rightarrow$: 1. Denotes material conditional, and is read as "implies". If $P$ and $Q$ are logical predicates, $P\Rightarrow Q$ means that if $P$ is true, then $Q$ is also true. Thus, $P\Rightarrow Q$ is logically equivalent with $Q\lor \neg P$ .; 2. Often used improperly^[3] in plain text as an abbreviation of "implies".
$\Leftrightarrow$: 1. Denotes logical equivalence, and is read "is equivalent to" or "if and only if". If $P$ and $Q$ are logical predicates, $P\Leftrightarrow Q$ is thus an abbreviation of $(P\Rightarrow Q)\land (Q\Rightarrow P)$ , or of $(P\land Q)\lor (\neg P\land \neg Q)$ .; 2. Often used improperly^[3] in plain text as an abbreviation of "if and only if".
$⊤$: 1. $\top$ denotes the logical predicate always true.; 2. Denotes also the truth value true.; 3. Sometimes denotes the top element of a bounded lattice (previous meanings are specific examples).; 4. For the use as a superscript, see $⊤ □$ .
$⊥$: 1. $\bot$ denotes the logical predicate always false.; 2. Denotes also the truth value false.; 3. Sometimes denotes the bottom element of a bounded lattice (previous meanings are specific examples).; 4. As a binary operator, denotes perpendicularity and orthogonality. For example, if $A, B, C$ are three points in a Euclidean space, then $AB\perp AC$ means that the line segments $AB$ and $AC$ are perpendicular, and form a right angle.; 5. In Cryptography often denotes an error in place of a regular value.; 6. For the use as a superscript, see $□ ⊥$ .

Blackboard bold[]

The blackboard bold typeface is widely used for denoting the basic number systems. These systems are often also denoted by the corresponding uppercase bold letter. A clear advantage of blackboard bold is that these symbols cannot be confused with anything else. This allows using them in any area of mathematics, without having to recall their definition. For example, if one encounters $\mathbb {R}$ in combinatorics, one should immediately know that this denotes the real numbers, although combinatorics does not study the real numbers (but it uses them for many proofs).

$\mathbb {N}$: Denotes the set of natural numbers $\{0,1,2,\ldots \}$ , or sometimes $\{1,2,\ldots \}$ . It is often denoted also by $\mathbf {N}$ .
$\mathbb {Z}$: Denotes the set of integers $\{\ldots ,-2,-1,0,1,2,\ldots \}$ . It is often denoted also by $\mathbf {Z}$ .
$\mathbb {Z} _{p}$: 1. Denotes the set of $p$ -adic integers, where $p$ is a prime number.; 2. Sometimes, $\mathbb {Z} _{n}$ denotes the integers modulo $n$ , where $n$ is an integer greater than 0. The notation $\mathbb {Z} /n\mathbb {Z}$ is also used, and is less ambiguous.
$\mathbb {Q}$: Denotes the set of rational numbers (fractions of two integers). It is often denoted also by $\mathbf {Q}$ .
$\mathbb {Q} _{p}$: Denotes the set of $p$ -adic numbers, where $p$ is a prime number.
$\mathbb {R}$: Denotes the set of real numbers. It is often denoted also by $\mathbf {R}$ .
$\mathbb {C}$: Denotes the set of complex numbers. It is often denoted also by $\mathbf {C}$ .
$\mathbb {H}$: Denotes the set of quaternions. It is often denoted also by $\mathbf {H}$ .
$\mathbb {F} _{q}$: Denotes the finite field with $q$ elements, where $q$ is a prime power (including prime numbers). It is denoted also by $GF(q)$ .

Calculus[]

$□'$: Lagrange's notation for the derivative: if $f$ is a function of a single variable, $f'$ , read as "f prime", is the derivative of $f$ with respect to this variable. The second derivative is the derivative of $f'$ , and is denoted $f''$ .
${\dot {\Box }}$: Newton's notation, most commonly used for the derivative with respect to time: if $x$ is a variable depending on time, then ${\dot {x}}$ is its derivative with respect to time. In particular, if $x$ represents a moving point, then ${\dot {x}}$ is its velocity.
${\ddot {\Box }}$: Newton's notation, for the second derivative: in particular, if $x$ is a variable that represents a moving point, then ${\ddot {x}}$ is its acceleration.
$.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}d□/d□$: Leibniz's notation for the derivative, which is used in several slightly different ways.; 1. If $y$ is a variable that depends on $x$ , then $\textstyle {\frac {dy}{dx}}$ , read as "d y over d x", is the derivative of $y$ with respect to $x$ .; 2. If $f$ is a function of a single variable $x$ , then $\textstyle {\frac {df}{dx}}$ is the derivative of $f$ , and $\textstyle {\frac {df}{dx}}(a)$ is the value of the derivative at $a$ .; 3. Total derivative: if $f(x_{1},\ldots ,x_{n})$ is a function of several variables that depend on $x$ , then $\textstyle {\frac {df}{dx}}$ is the derivative of $f$ considered as a function of $x$ . That is, $\textstyle {\frac {df}{dx}}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}\,{\frac {dx_{i}}{dx}}$ .
$\partial□ / \partial□$: Partial derivative: if $f(x_{1},\ldots ,x_{n})$ is a function of several variables, $\textstyle {\frac {\partial f}{\partial x_{i}}}$ is the derivative with respect to the $i$ th variable considered as an independent variable, the other variables being considered as constants.

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