Editing Glossary of mathematical symbols

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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 [[symbol]]s that occur in a [[mathematical formula|formula]] or a [[mathematical expression]]. More formally, a ''mathematical symbol'' is any [[grapheme]] used in mathematical formulas and expressions. As formulas and expressions are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

The most basic symbols are the [[decimal digit]]s (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 [[point (geometry)|points]] in geometry, and lower-case letters were used for [[variable (mathematics)|variables]] and [[constant (mathematics)|constants]]. Letters are used for representing many other types of [[mathematical object]]. As the number of these types has increased, the [[Greek alphabet]] and some [[Hebrew alphabet|Hebrew letters]] have also come to be used. For more symbols, other typefaces are also used, mainly [[boldface]] {{tmath|1= \mathbf {a,A,b,B},\ldots }}, [[script typeface]] <math>\mathcal {A,B},\ldots</math> (the lower-case script face is rarely used because of the possible confusion with the standard face), [[Fraktur|German fraktur]] {{tmath|1= \mathfrak {a,A,b,B},\ldots }}, and [[blackboard bold]] {{tmath|1= \mathbb{N, Z, Q, R, C, H, F}_q }} (the other letters are rarely used in this face, or their use is unconventional).  It is commonplace to use alphabets, fonts and typefaces to group symbols by type (for example, boldface is often used for [[vector (mathematics and physics)|vectors]] and uppercase for [[matrix (mathematics)|matrices]]). <!-- Donald Knuth's TeXbook describes a convention where, in mathematical [[formula]]s, the standard [[typeface]] is [[italic type]] for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. ISO/IEC 80000-1 gives another convention in which italics is used for variables and upright (roman) font for constants, for both Latin and Greek letters. -->

The use of specific Latin and Greek letters as symbols for denoting mathematical objects is not described in this article. For such uses, see ''[[Variable (mathematics)#Conventional variable names|Variable § Conventional variable names]]'' 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 <math>\textstyle\prod{}</math> and <math>\textstyle\sum{}</math>.

These letters alone are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin in [[punctuation mark]]s and [[diacritic]]s traditionally used in [[typography]]; others by deforming [[letter form]]s, as in the cases of <math>\in</math> and <math>\forall</math>. Others, such as {{math|+}} and {{math|1==}}, were specially designed for mathematics.

== Layout of this article ==

* 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 [[#Brackets|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 the symbol that they are looking for is related, 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 <math>\Box</math> is used for representing the neighboring parts of a formula that contains the symbol. See {{slink||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 engine]]s and [[copy and paste|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 [[WP:ANCHOR|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 ==
{{glossary}}
{{term|1=+|content= {{math|1=+}} {{spaces|3|em}}([[plus sign]])}}
{{defn |no=1|1=Denotes [[addition]] and is read as ''plus''; for example, {{math|3 + 2}}.}}
{{defn |no=2 |1=Denotes that a number is [[Sign_(mathematics)#Terminology_for_signs|positive]] and is read as ''plus''. Redundant, but sometimes used for emphasizing that a number is [[positive number|positive]], specially when other numbers in the context are or may be negative; for example, {{math|+2}}.}}
{{defn |no=3 |1=Sometimes used instead of <math>\sqcup</math> for a [[disjoint union]] of [[set (mathematics)|sets]].}}

{{term|1=−|content= {{math|1=−}} {{spaces|3|em}}([[minus sign]])}}
{{defn |no=1 |1=Denotes [[subtraction]] and is read as ''minus''; for example, {{math|3 − 2}}.}}
{{defn |no=2 |1=Denotes the [[additive inverse]] and is read as ''minus'', '' the negative of'', or ''the opposite of''; for example, {{math|−2}}.}}
{{defn|no=3|1= Also used in place of {{math|\}} for denoting the [[set-theoretic complement]]; see [[#∖|\]] in {{slink||Set theory}}.}}

{{term|1=×|content= {{math|1=×}} {{spaces|3|em}}([[multiplication sign]])}}
{{defn |no=1|1=In [[elementary arithmetic]], denotes [[multiplication]], and is read as ''times''; for example, {{math|3 × 2}}.}}
{{defn |no=2|1= In [[geometry]] and [[linear algebra]], denotes the [[cross product]].}}
{{defn|no=3|1=In [[set theory]] and [[category theory]], denotes the [[Cartesian product]] and the [[direct product]]. See also [[#cartesian|×]] in {{slink||Set theory}}.}}

{{term|1=·|content= {{math|1=·}} {{spaces|3|em}}([[interpunct|dot]])}}
{{defn |no=1|1=Denotes [[multiplication]] and is read as ''times''; for example, {{math|3 ⋅ 2}}.}}
{{defn |no=2|1= In [[geometry]] and [[linear algebra]], denotes the [[dot product]].}}
{{defn |no=3|1= Placeholder used for replacing an indeterminate element. For example, saying "the [[absolute value]] is denoted by {{math|{{abs}}}}" is perhaps clearer than saying that it is denoted as {{math|{{!}} {{!}}}}.}}

{{term|1=±|content= {{math|1=±}} {{spaces|3|em}}([[plus–minus sign]])}}
{{defn|no=1|1=Denotes either a plus sign or a minus sign.}}
{{defn|no=2|1=Denotes the range of values that a measured quantity may have; for example, {{math|10 ± 2}} denotes an unknown value that lies between 8 and 12.}}

{{term|1=∓|content= {{math|1=∓}} {{spaces|3|em}}([[minus-plus sign]])}}
{{defn|1=Used paired with {{math|±}}, denotes the opposite sign; that is, {{math|+}} if {{math|±}} is {{math|−}}, and {{math|−}} if {{math|±}} is {{math|+}}.}}

{{term|1=÷|content= {{math|1=÷}} {{spaces|3|em}}([[division sign]])}}
{{defn|1=Widely used for denoting [[division (mathematics)|division]] in [[English-speaking world| Anglophone]] countries, it is no longer in common use in mathematics and its use is "not recommended".<ref name=ISO>[[ISO 80000-2]], Section 9 "Operations", 2-9.6</ref> In some countries, it can indicate subtraction.}}

{{term|ratio|content= {{math|1=:}} {{spaces|3|em}}([[colon (punctuation)|colon]])}}
{{defn|no=1|Denotes the [[ratio]] of two quantities.}}
{{defn|no=2|In some countries, may denote [[division (mathematics)|division]].}}
{{defn|no=3|In [[set-builder notation]], it is used as a separator meaning "such that"; see [[#b:b|{{math|{{mset|□ : □}}}}]].}}

{{term|/|content= {{math|1=/}} {{spaces|3|em}}([[slash (punctuation)|slash]])}}
{{defn|no=1|1=Denotes [[division (mathematics)|division]] and is read as ''divided by'' or ''over''. Often replaced by a horizontal bar. For example, {{math|3 / 2}} or <math>\frac 32</math>.}}
{{defn|no=2|1=Denotes a [[quotient (disambiguation)#Mathematics|quotient structure]]. For example, [[quotient set]], [[quotient group]], [[quotient category]], etc.}}
{{defn|no=3|In [[number theory]] and [[Field theory (mathematics)|field theory]], <math>F/E</math> denotes a [[field extension]], where {{mvar|F}} is an [[extension field]] of the [[field (mathematics)|field]] {{mvar|E}}.}}
{{defn|no=4|1=In [[probability theory]], denotes a [[conditional probability]]. For example, <math>P(A/B)</math> denotes the probability of {{mvar|A}}, given that {{mvar|B}} occurs. Usually denoted <math>P(A\mid B)</math>: see "[[#vbar|{{math|{{!}}}}]]".}}

{{term|√|content= {{math|1=√}} {{spaces|3|em}}([[radical symbol#Encoding|square-root symbol]])}}
{{defn|Denotes [[square root]] and is read as ''the square root of''. Rarely used in modern mathematics without a horizontal bar delimiting the width of its argument (see the next item). For example, {{math|√2}}.}}

{{term|sqrt|content= {{math|{{sqrt|&nbsp;&nbsp;}}}} {{spaces|2|em}}([[radical symbol]])}}
{{defn|no=1|1=Denotes [[square root]] and is read as ''the square root of''. For example, <math>\sqrt{3+2}</math>.}}
{{defn|no=2|1=With an integer greater than 2 as a left superscript, denotes an [[nth root|{{mvar|n}}th root]]. For example, <math>\sqrt[7]{3}</math> denotes the 7th root of 3.}}

{{term|caret|content= {{math|1=^}} {{spaces|3|em}}([[caret]])}}
{{defn|no=1|[[Exponentiation]] is normally denoted with a [[superscript]]. However, <math>x^y</math> is often denoted {{math|''x''^''y''}} when superscripts are not easily available, such as in [[programming language]]s (including [[LaTeX]]) or plain text [[email]]s.}}
{{defn|no=2|Not to be confused with [[#∧|∧]]}}
{{glossary end}}

== Equality, equivalence and similarity ==
{{glossary}}
{{term|equal|content= {{math|1==}} {{spaces|3|em}}([[equals sign]])}}
{{defn|no=1|Denotes [[equality (mathematics)|equality]].}}
{{defn|no=2|Used for naming a [[mathematical object]] in a sentence like "let <math>x=E</math>{{space|hair}}", where {{mvar|E}} is an [[expression (mathematics)|expression]]. See also {{math|≝}}, {{math|≜}} {{nowrap|or <math>:=</math>.}}
}}

{{term|≝|content= <math>\triangleq \quad \stackrel{\scriptscriptstyle \mathrm{def}}{=} \quad  :=</math>}}
{{defn|Any of these is sometimes used for naming a [[mathematical object]]. Thus, <math>x \triangleq E,</math>  <math>x\mathrel{\stackrel{\scriptscriptstyle \mathrm{def}}{=}}E,</math> <math>x \mathrel{:=} E</math> and <math>E \mathrel{=:} x</math> are each an abbreviation of the phrase "let <math>x = E</math>", where {{tmath|E}} is an [[expression (mathematics)|expression]] and {{tmath|x}} is a [[variable (mathematics)|variable]].
This is similar to the concept of [[assignment (computer science)|assignment]] in computer science, which is variously denoted (depending on the [[programming language]] used) <math>=, :=, \leftarrow, \ldots</math>}}

{{term|inequal|content= {{math|1=≠}} {{spaces|3|em}}([[not-equal sign]])}}
{{defn|Denotes [[inequality (mathematics)|inequality]] and means "not equal".}}

{{term|approx|content= {{math|1=≈}} }}
{{defn|The most common symbol for denoting [[approximate equality]]. For example, <math>\pi \approx 3.14159.</math> }}

{{term|tilde|content= {{math|1=~}} {{spaces|3|em}}([[tilde]])}}
{{defn|no=1|1=Between two numbers, either it is used instead of {{math|≈}} to mean "approximatively equal", or it means "has the same [[order of magnitude]] as".}}
{{defn|no=2|Denotes the [[asymptotic equivalence]] of two functions or sequences.}}
{{defn|no=3|1=Often used for denoting other types of similarity, for example, [[matrix similarity]] or [[similarity (geometry)|similarity of geometric shapes]].}}
{{defn|no=4|Standard notation for an [[equivalence relation]].}}
{{defn|no=5|In [[probability]] and [[statistics]], may specify the [[probability distribution]] of a [[random variable]]. For example, <math>X\sim N(0,1)</math> means that the distribution of the random variable {{mvar|X}} is  [[standard normal distribution|standard normal]].<ref>{{Cite book|url=https://archive.org/details/statisticsdataan0000tamh|title=Statistics and Data Analysis: From Elementary to Intermediate|date=2000 |isbn=978-0-13-744426-7 }}</ref>}}
{{defn|no=6|Notation for [[Proportionality_(mathematics)|proportionality]]. See also [[#∝|{{math|∝}}]] for a less ambiguous symbol.}}

{{term|triple bar|content= {{math|1=≡}} {{spaces|3|em}}([[triple bar]])}}
{{defn|no=1|Denotes an [[identity (mathematics)|identity]]; that is, an equality that is true whichever values are given to the variables occurring in it.}}
{{defn|no=2|In [[number theory]], and more specifically in [[modular arithmetic]], denotes the [[congruence modulo n|congruence]] modulo an integer.}}
{{defn|no=3|May denote a [[logical equivalence]].}}

{{term|1=≅|content= <math>\cong</math>}}
{{defn|no=1|May denote an [[isomorphism]] between two [[mathematical structures]], and is read as "is isomorphic to".}}
{{defn|no=2|In [[geometry]], may denote the [[congruence (geometry)|congruence]] of two [[geometric shape]]s (that is the equality [[up to]] a [[displacement (geometry)|displacement]]), and is read "is congruent to".}}
{{glossary end}}

== Comparison ==
{{glossary}}
{{term|less|content= {{math|1=<}} {{spaces|3|em}}([[less-than sign]])}}
{{defn|no=1|[[Strict inequality]] between two numbers; means and is read as "[[less than]]".}}
{{defn|no=2|Commonly used for denoting any [[strict order]].}}
{{defn|no=3|Between two [[group (mathematics)|groups]], may mean that the first one is a [[proper subgroup]] of the second one.}}

{{term|greater|content= {{math|1=>}} {{spaces|3|em}}([[greater-than sign]])}}
{{defn|no=1|[[Strict inequality]] between two numbers; means and is read as "[[greater than]]".}}
{{defn|no=2|Commonly used for denoting any [[strict order]].}}
{{defn|no=3|Between two [[group (mathematics)|groups]], may mean that the second one is a [[proper subgroup]] of the first one.}}

{{term|less-equal sign|content= {{math|1=≤}} }}
{{defn|no=1|Means "[[less than or equal to]]". That is, whatever {{mvar|A}} and {{mvar|B}} are, {{math|''A'' ≤ ''B''}} is equivalent to {{math|''A'' < ''B'' or ''A'' {{=}} ''B''}}.}}
{{defn|no=2|Between two [[group (mathematics)|groups]], may mean that the first one is a [[subgroup]] of the second one.}}

{{term|greater-equal sign|content= {{math|1=≥}} }}
{{defn|no=1|Means "[[greater than or equal to]]". That is, whatever {{mvar|A}} and {{mvar|B}} are, {{math|''A'' ≥ ''B''}} is equivalent to {{math|''A'' > ''B'' or ''A'' {{=}} ''B''}}.}}
{{defn|no=2|Between two [[group (mathematics)|groups]], may mean that the second one is a [[subgroup]] of the first one.}}

{{term|much-greater-or-less|content= <math>\ll \text{ and }\gg</math>}} 
{{defn|no=1|Means "[[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]].}}
{{defn|no=2|In [[measure theory]], <math>\mu\ll\nu</math> means that the measure <math>\mu</math> is absolutely continuous with respect to the measure <math>\nu</math>.}}

{{term|less-equal sign|content= <math>\leqq</math>}}
{{defn|A rarely used symbol, generally a synonym of {{math|≤}}.}}

{{term|pred-succ|content= <math>\prec \text{ and } \succ</math>}} 
{{defn|no=1|Often used for denoting an [[partial order|order]] or, more generally, a [[preorder]], when it would be confusing or not convenient to use {{math|<}} and {{math|>}}.}}
{{defn|no=2|[[Sequention]] in [[asynchronous logic]].}}
{{glossary end}}

== Set theory ==
{{glossary}}
{{term|∅|content= {{math|∅}}}}
{{defn|Denotes the [[empty set]], and is more often written <math>\emptyset</math>. Using [[set-builder notation]], it may also be denoted [[#bb|<math>\{\}</math>]].}}

{{term|sharp|content= {{math|#}} {{spaces|3|em}}([[number sign]])}}
{{defn|term=sharp|no=1
|defn=Number of elements: <math>\#{}S</math> may denote the [[cardinality]] of the [[set (mathematics)|set]] {{mvar|S}}. An alternative notation is <math>|S|</math>; see [[#!□!|<math>|\square|</math>]].}}
{{defn|term=sharp|no=2
|defn=[[Primorial]]: <math>n{}\#</math> denotes the product of the [[prime number|prime numbers]] that are not greater than {{mvar|n}}.}}
{{defn|term=sharp|no=3
|defn=In [[topology]], <math>M\#N</math> denotes the [[connected sum]] of two [[manifolds]] or two [[knot (mathematics)|knots]].}}

{{term|∈|content= {{math|∈}}}}
{{defn|Denotes [[set membership]], and is read "is in", "belongs to", or "is a member of". That is, <math>x\in S</math> means that {{mvar|x}} is an element of the set {{mvar|S}}.}}

{{term|∉|content= {{math|∉}}}}
{{defn|Means "is not in". That is, <math>x\notin S</math> means <math>\neg(x\in S)</math>.}}

{{term|⊂|content= {{math|⊂}}}}
{{defn|Denotes [[set inclusion]]. However two slightly different definitions are common.}}
{{defn|term=⊂|no=1
|defn=<math>A\subset B</math> may mean that {{mvar|A}} is a [[subset]] of {{mvar|B}}, and is possibly equal to {{mvar|B}}; that is, every element of {{mvar|A}} belongs to {{mvar|B}}; expressed as a formula, <math>\forall{}x, \,x\in A \Rightarrow x\in B</math>.}}
{{defn|term=⊂|no=2
|defn=<math>A\subset B</math> may mean that {{mvar|A}} is a [[proper subset]] of {{mvar|B}}, that is the two sets are different, and every element of {{mvar|A}} belongs to {{mvar|B}}; expressed as a formula, <math>A\neq B \land\forall{}x, \,x\in A \Rightarrow x\in B</math>.}}

{{term|⊆|content= {{math|⊆}}}}
{{defn|<math>A\subseteq B</math> means that {{mvar|A}} is a [[subset]] of {{mvar|B}}. Used for emphasizing that equality is possible, or when {{gli|⊂-defn2|<math>A\subset B</math>}} means that <math>A</math> is a proper subset of <math>B.</math>}}

{{term|⊊|content= {{math|⊊}}}}
{{defn|<math>A\subsetneq B</math> means that {{mvar|A}} is a [[proper subset]] of {{mvar|B}}. Used for emphasizing that <math>A\neq B</math>, or when {{gli|⊂-defn1|<math>A\subset B</math>}} does not imply that <math>A</math> is a proper subset of <math>B.</math>}}

{{term|term=⊃|content={{math|⊃, ⊇, ⊋}}}}
{{defn|Denote the converse relation of {{gli|⊂|<math>\subset</math>}}, {{gli|⊆|<math>\subseteq</math>}}, and {{gli|⊊|<math>\subsetneq</math>}} respectively. For example, <math>B\supset A</math> is equivalent to <math>A\subset B</math>.}}

{{term|∪|content= {{math|∪}}}}
{{defn|Denotes [[set-theoretic union]], that is, <math>A\cup B</math> is the set formed by the elements of {{mvar|A}} and {{mvar|B}} together. That is, <math>A\cup B=\{x\mid (x\in A) \lor (x\in B)\}</math>.}}

{{term|∩|content= {{math|∩}}}}
{{defn|Denotes [[set-theoretic intersection]], that is, <math>A\cap B</math> is the set formed by the elements of both {{mvar|A}} and {{mvar|B}}. That is, <math>A\cap B=\{x\mid (x\in A) \land (x\in B)\}</math>.}}

{{term|∖|content= {{math|∖}} {{spaces|3|em}}([[backslash]])}}
{{defn|[[Set difference]]; that is, <math>A\setminus B</math> is the set formed by the elements of {{mvar|A}} that are not in {{mvar|B}}. Sometimes, <math>A-B</math> is used instead; see [[#−|−]] in {{slink||Arithmetic operators}}.}}

{{term|⊖|{{math|⊖}} or <math>\triangle</math>}}
{{defn|[[Symmetric difference]]: that is, <math>A\ominus B</math> or <math>A\operatorname{\triangle}B</math> is the set formed by the elements that belong to exactly one of the two sets {{mvar|A}} and {{mvar|B}}.}}

{{term|∁|content=<math>\complement</math> }}
{{defn|no=1|With a subscript, denotes a [[set complement]]: that is, if <math>B\subseteq A</math>, then <math>\complement_A B = A\setminus B</math>.}}
{{defn|no=2|Without a subscript, denotes the [[absolute complement]]; that is, <math>\complement A  = \complement_ U A</math>, where {{mvar|U}} is a set implicitly defined by the context, which contains all sets under consideration. This set {{mvar|U}} is sometimes called the [[universe of discourse]]. }}

{{term|cartesian|content= {{math|×}} {{spaces|3|em}}([[multiplication sign]])}}
{{defn|See also [[#×|×]] in {{slink||Arithmetic operators}}.}}
{{defn|no=1|Denotes the [[Cartesian product]] of two sets. That is, <math>A\times B</math> is the set formed by all [[pair (mathematics)|pairs]] of an element of {{mvar|A}} and an element of {{mvar|B}}.}}
{{defn|no=2|Denotes the [[direct product]] of two [[mathematical structure]]s 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]], [[product topology|direct product of topological spaces]].}}
{{defn|no=3|In [[category theory]], denotes the [[product (category theory)|direct product]] (often called simply ''product'') of two objects, which is a generalization of the preceding concepts of product.}}

{{term|⊔|content= <math>\sqcup
</math> }}
{{defn|Denotes the [[disjoint union]]. That is, if {{mvar|A}} and {{mvar|B}} are sets then <math>A\sqcup B=\left(A\times\{i_A\}\right)\cup\left(B\times\{i_B\}\right)</math> is a set of [[ordered pair|pairs]] where {{mvar|i<sub>A</sub>}} and {{mvar|i<sub>B</sub>}} are distinct indices discriminating the members of {{mvar|A}} and {{mvar|B}} in {{tmath|A\sqcup B}}.}}

{{term|∐|content=<math>\bigsqcup \text{ or  } \coprod</math> }}
{{defn|no=1|Used for the [[disjoint union]] of a family of sets, such as in <math display=inline>\bigsqcup_{i\in I}A_i.</math>}}
{{defn|no=2|Denotes the [[coproduct]] of [[mathematical structure]]s or of objects in a [[category (mathematics)|category]].}}

{{glossary end}}

== Basic logic ==
Several [[logical symbol]]s 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]]''.

{{glossary}}
{{term|¬|content= {{math|¬}} {{spaces|3|em}}([[not sign]])}}
{{defn|Denotes [[logical negation]], and is read as "not". If {{mvar|E}} is a [[logical predicate]], <math>\neg E</math> is the predicate that evaluates to ''true'' if and only if {{mvar|E}} evaluates to ''false''. For clarity, it is often replaced by the word "not". In [[programming language]]s and some mathematical texts, it is sometimes replaced by "{{math|~}}" or "{{math|!}}", which are easier to type on some keyboards.}}

{{term|∨|content= {{math|∨}} {{spaces|3|em}}([[descending wedge]])}}
{{defn|no=1|Denotes the [[logical or]], and is read as "or". If {{mvar|E}} and {{mvar|F}} are [[logical predicate]]s, <math>E\lor F</math> is true if either {{mvar|E}}, {{mvar|F}}, or both are true. It is often replaced by the word "or".}}
{{defn|no=2|In [[lattice theory]], denotes the [[join (lattice theory)|join]] or [[least upper bound]] operation.}} 
{{defn|no=3|In [[topology]], denotes the [[wedge sum]] of two [[pointed space]]s.}}

{{term|∧|content= {{math|∧}} {{spaces|3|em}}([[wedge (symbol)|wedge]])}}
{{defn|no=1|Denotes the [[logical and]], and is read as "and". If {{mvar|E}} and {{mvar|F}} are [[logical predicate]]s, <math>E\land F</math> is true if {{mvar|E}} and {{mvar|F}} are both true. It is often replaced by the word "and" or the symbol "{{math|&}}".}}
{{defn|no=2|In [[lattice theory]], denotes the [[meet (lattice theory)|meet]] or [[greatest lower bound]] operation.}} 
{{defn|no=3|In [[multilinear algebra]], [[geometry]], and [[multivariable calculus]], denotes the [[wedge product]] or the [[exterior product]].}}

{{term|⊻|{{math|⊻}}}}
{{defn|[[Exclusive or]]: if {{mvar|E}} and {{mvar|F}} are two [[Boolean variable]]s or [[predicate (mathematical logic)#Simplified overview|predicate]]s, <math>E\veebar F</math> denotes the exclusive or. Notations {{math|''E'' <small>'''XOR'''</small> ''F''}} and <math>E\oplus F</math> are also commonly used; see [[#⊕|⊕]].}}

{{term|∀|content= {{math|∀}} {{spaces|3|em}}([[turned A]])}}
{{defn|no=1|Denotes [[universal quantification]] and is read as "for all". If {{mvar|E}} is a [[logical predicate]], <math>\forall x \; E</math> means that {{mvar|E}} is true for all possible values of the variable {{mvar|x}}.}}
{{defn|no=2|Often used in plain text as an abbreviation of "for all" or "for every".}}

{{term|∃|content={{math|∃}}}}
{{defn|no=1|Denotes [[existential quantification]] and is read "there exists ... such that". If {{mvar|E}} is a [[logical predicate]], <math>\exists x \; E</math> means that there exists at least one value of {{mvar|x}} for which {{mvar|E}} is true.}}
{{defn|no=2|Often used in plain text as an abbreviation of "there exists".}}

{{term|∃!|content={{math|∃!}}}}
{{defn|Denotes [[uniqueness quantification]], that is, <math>\exists ! x \; P</math> means "there exists exactly one {{mvar|x}} such that {{mvar|P}} (is true)". In other words,
<math>\exists ! x \; P(x)</math> is an abbreviation of <math>\exists x\,( P(x) \, \wedge \neg \exists y\,(P(y) \wedge y  \ne x))</math>.}}

{{term|⇒|content={{math|⇒}}}}
{{defn|no=1|Denotes [[material conditional]], and is read as "implies". If {{mvar|P}} and {{mvar|Q}} are [[logical predicate]]s, <math>P \Rightarrow Q</math> means that if {{mvar|P}} is true, then {{mvar|Q}} is also true. Thus, <math>P \Rightarrow Q</math> is logically equivalent with <math>Q\lor \neg P</math>.}}
{{defn|no=2|Often used in plain text as an abbreviation of "implies".}}

{{term|⇔|content={{math|⇔}}}}
{{defn|no=1|Denotes [[logical equivalence]], and is read "is equivalent to" or "[[if and only if]]". If  {{mvar|P}} and {{mvar|Q}} are [[logical predicate]]s, <math>P \Leftrightarrow Q</math> is thus an abbreviation of <math>(P \Rightarrow Q) \land (Q \Rightarrow P)</math>, or of <math>(P \land Q) \lor (\neg P \land \neg Q)</math>.}}
{{defn|no=2|Often used in plain text as an abbreviation of "[[if and only if]]".}}

{{term|⊤|content= {{math|⊤}} {{spaces|3|em}}([[Tee (symbol)|tee]])}}
{{defn|no=1|<math>\top</math> denotes the [[logical predicate]] ''always true''.}}
{{defn|no=2|Denotes also the [[truth value]] ''true''.}}
{{defn|no=3|Sometimes denotes the [[top element]] of a [[bounded lattice]] (previous meanings are specific examples).}}
{{defn|no=4|For the use as a superscript, see [[#□⊤|{{math|□{{sup|⊤}}}}]].}}

{{term|⊥|content= {{math|⊥}} {{spaces|3|em}}([[up tack]])}}
{{defn|no=1|<math>\bot</math> denotes the [[logical predicate]] ''always false''.}}
{{defn|no=2|Denotes also the [[truth value]] ''false''.}}
{{defn|no=3|Sometimes denotes the [[bottom element]] of a [[bounded lattice]] (previous meanings are specific examples).}}
{{defn|no=4|In [[cryptography]] often denotes an error in place of a regular value.}}
{{defn|no=5|For the use as a superscript, see [[#□⊥|{{math|□{{sup|⊥}}}}]].}}
{{defn|no=6|For the similar symbol, see [[#⟂|<math>\perp</math>]].}}

{{glossary end}}

== Blackboard bold ==

The [[blackboard bold]] [[typeface]] is widely used for denoting the basic [[number system]]s. 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 <math>\mathbb R</math> in [[combinatorics]], one should immediately know that this denotes the [[real number]]s, although combinatorics does not study the real numbers (but it uses them for many proofs).
{{glossary}}
{{term|ℕ|content=<math>\mathbb N</math>}}
{{defn|Denotes the set of [[natural number]]s <math>\{1, 2,\ldots \},</math> or sometimes <math>\{0, 1, 2, \ldots \}.</math> When the distinction is important and readers might assume either definition, <math>\mathbb{N}_1</math> and <math>\mathbb{N}_0</math> are used, respectively, to denote one of them unambiguously. Notation <math>\mathbf N</math> is also commonly used.}}

{{term|ℤ|content=<math>\mathbb Z</math>}}
{{defn|Denotes the set of [[integer]]s <math>\{\ldots, -2, -1, 0, 1, 2,\ldots \}.</math> It is often denoted also by <math>\mathbf Z.</math>}}

{{term|ℤp|content=<math>\mathbb{Z}_p</math>}}
{{defn|no=1|Denotes the set of [[p-adic integer|{{mvar|p}}-adic integers]], where {{mvar|p}} is a [[prime number]].}}
{{defn|no=2|Sometimes, <math>\mathbb Z_n</math> denotes the [[integers modulo n|integers modulo {{mvar|n}}]], where {{mvar|n}} is an [[integer]] greater than 0. The notation <math>\mathbb Z/n\mathbb Z</math> is also used, and is less ambiguous.}}

{{term|ℚ|content=<math>\mathbb Q</math>}}
{{defn|Denotes the set of [[rational number]]s (fractions of two integers). It is often denoted also by <math>\mathbf Q.</math>}}

{{term|ℚp|content=<math>\mathbb{Q}_p</math>}}
{{defn|Denotes the set of [[p-adic number|{{mvar|p}}-adic numbers]], where {{mvar|p}} is a [[prime number]].}}

{{term|ℝ|content=<math>\mathbb R</math>}}
{{defn|Denotes the set of [[real number]]s. It is often denoted also by <math>\mathbf R.</math>}}

{{term|ℂ|content=<math>\mathbb C</math>}}
{{defn|Denotes the set of [[complex number]]s. It is often denoted also by <math>\mathbf C.</math>}}

{{term|ℍ|content=<math>\mathbb H</math>}}
{{defn|Denotes the set of [[quaternion]]s. It is often denoted also by <math>\mathbf H.</math>}}

{{term|Fq|content=<math>\mathbb{F}_q</math>}}
{{defn|Denotes the [[finite field]] with {{mvar|q}} elements, where {{mvar|q}} is a [[prime power]] (including [[prime number]]s). It is denoted also by {{math|GF(''q'')}}.}}

{{term|O|content=<math>\mathbb O</math>}}
{{defn|Used on rare occasions to denote the set of [[octonion]]s. It is often denoted also by <math>\mathbf O.</math>}}

{{glossary end}}

== Calculus ==
{{glossary}}
{{term|□'|{{math|□{{'}}}}}}
{{defn|[[Lagrange's notation]] for the [[derivative]]: If {{mvar|f}} is a [[function (mathematics)|function]] of a single variable, <math>f'</math>, read as "f [[Prime_(symbol)#Use_in_mathematics,_statistics,_and_science|prime]]", is the derivative of {{mvar|f}} with respect to this variable. The [[second derivative]] is the derivative of <math>f'</math>, and is denoted <math>f''</math>.}}

{{term|\dot|<math>\dot \Box</math>}}
{{defn|[[Newton's notation]], most commonly used for the [[derivative]] with respect to time. If {{mvar|x}} is a variable depending on time, then <math>\dot x,</math> read as "x dot", is its derivative with respect to time. In particular, if {{mvar|x}} represents a moving point, then <math>\dot x</math> is its [[velocity]].}}

{{term|\ddot|<math>\ddot \Box</math>}}
{{defn|[[Newton's notation]], for the [[second derivative]]: If {{mvar|x}} is a variable that represents a moving point, then <math>\ddot x</math> is its [[acceleration]].}}

{{term|Leibnitz|{{math|{{sfrac|d □|d □}}}}}}
{{defn|[[Leibniz's notation]] for the [[derivative]], which is used in several slightly different ways.}}
{{defn|no=1|If {{mvar|y}} is a variable that [[dependent variable|depends]] on {{mvar|x}}, then <math>\textstyle \frac{\mathrm{d}y}{\mathrm{d}x}</math>, read as "d y over d x" (commonly shortened to "d y d x"), is the derivative of {{mvar|y}} with respect to {{mvar|x}}.}}
{{defn|no=2|If {{mvar|f}} is a [[function (mathematics)|function]] of a single variable {{mvar|x}}, then <math>\textstyle \frac{\mathrm{d}f}{\mathrm{d}x}</math> is the derivative of {{mvar|f}}, and 
<math>\textstyle \frac{\mathrm{d}f}{\mathrm{d}x}(a)</math> is the value of the derivative at {{mvar|a}}.}}
{{defn|no=3|[[Total derivative]]: If <math>f(x_1, \ldots, x_n)</math> is a [[function (mathematics)|function]] of several variables that [[dependent variable|depend]] on {{mvar|x}}, then <math>\textstyle \frac{\mathrm{d}f}{\mathrm{d}x}</math> is the derivative of {{mvar|f}} considered as a function of {{mvar|x}}. That is, <math>\textstyle \frac{\mathrm{d}f}{dx}=\sum_{i=1}^n \frac{\partial f}{\partial x_i}\,\frac{\mathrm{d}x_i}{\mathrm{d}x}</math>.}}

{{term|partial|{{math|{{sfrac|∂ □|∂ □}}}}}}
{{defn|[[Partial derivative]]: If <math>f(x_1, \ldots, x_n)</math> is a [[function (mathematics)|function]] of several variables, <math>\textstyle\frac{\partial f}{\partial x_i}</math> is the derivative with respect to the {{mvar|i}}th variable considered as an [[independent variable]], the other variables being considered as constants.}}

{{term|functional|{{math|{{sfrac|𝛿 □|𝛿 □}}}}}}
{{defn|[[Functional derivative]]: If <math>f(y_1, \ldots, y_n)</math> is a [[functional (mathematics)|functional]] of several [[function (mathematics)|functions]], <math>\textstyle \frac{\delta f}{\delta y_{i}}</math> is the functional derivative with respect to the {{mvar|n}}th function considered as an [[independent variable]], the other functions being considered constant.}}

{{term|overline|<math>\overline\Box</math>}}
{{defn|no=1|[[Complex conjugate]]: If {{mvar|z}} is a [[complex number]], then <math>\overline{z}</math> is its complex conjugate. For example, <math>\overline{a+bi} = a-bi</math>.}}
{{defn|no=2|[[Topological closure]]: If {{mvar|S}} is a [[subset]] of a [[topological space]] {{mvar|T}}, then <math>\overline{S}</math> is its topological closure, that is, the smallest [[closed subset]] of {{mvar|T}} that contains {{mvar|S}}.}}
{{defn|no=3|[[Algebraic closure]]: If {{mvar|F}} is a [[field (mathematics)|field]], then <math>\overline{F}</math> is its algebraic closure, that is, the smallest [[algebraically closed field]] that contains {{mvar|F}}. For example, <math>\overline\mathbb Q</math> is the field of all [[algebraic number]]s.}}
{{defn|no=4|[[Mean value]]: If {{mvar|x}} is a [[variable (mathematics)|variable]] that takes its values in some sequence of numbers {{mvar|S}}, then <math>\overline{x}</math> may denote the mean of the elements of {{mvar|S}}.}}
{{defn|no=5|[[Negation]]: Sometimes used to denote negation of the entire expression under the bar, particularly when dealing with [[Boolean algebra]]. For example, one of [[De Morgan's laws]] says that <math>\overline{A \land B} = \overline{A} \lor \overline{B}</math> .}}

{{term|→|content={{math|→}}}}
{{defn|no=1|<math>A\to B</math> denotes a [[function (mathematics)|function]] with [[domain of a function|domain]] {{mvar|A}} and [[codomain]] {{mvar|B}}. For naming such a function, one writes <math>f:A \to B</math>, which is read as "{{mvar|f}} from {{mvar|A}} to {{mvar|B}}".}}
{{defn|no=2|More generally, <math>A\to B</math> denotes a [[homomorphism]] or a [[morphism]] from {{mvar|A}} to {{mvar|B}}.}} 
{{defn|no=3|May denote a [[logical implication]]. For the [[material conditional|material implication]] that is widely used in mathematics reasoning, it is nowadays generally replaced by [[#⇒|⇒]]. In [[mathematical logic]], it remains used for denoting implication, but its exact meaning depends on the specific theory that is studied.}}
{{defn|no=4|Over a [[variable (mathematics)|variable name]], means that the variable represents a [[vector (mathematics and physics)|vector]], in a context where ordinary variables represent [[scalar (mathematics)|scalar]]s; for example, <math>\overrightarrow v</math>. Boldface (<math>\mathbf v</math>) or a [[circumflex]] (<math>\hat v</math>) are often used for the same purpose.}}
{{defn|no=5|In [[Euclidean geometry]] and more generally in [[affine geometry]], <math>\overrightarrow{PQ}</math> denotes the [[vector (mathematics and physics)|vector]] defined by the two points {{mvar|P}} and {{mvar|Q}}, which can be identified with the [[Translation (geometry)|translation]] that maps {{mvar|P}} to {{mvar|Q}}. The same vector can be denoted also {{tmath|1= Q-P }}; see ''[[Affine space]]''.}}

{{term|↦|content={{math|↦}}}}
{{defn|"[[Maps to]]": Used for defining a [[function (mathematics)|function]] without having to name it. For example, <math>x\mapsto x^2</math> is the [[square function]].}}

{{term|∘|{{math|○}}<ref>The [[LaTeX]] equivalent to both [[Unicode]] symbols ∘ and ○ is \circ. The Unicode symbol that has the same size as \circ depends on the browser and its implementation. In some cases ∘ is so small that it can be confused with an [[interpoint]], and ○ looks similar as \circ. In other cases, ○ is too large for denoting a binary operation, and it is ∘ that looks like \circ. As LaTeX is commonly considered as the standard for mathematical typography, and it does not distinguish these two Unicode symbols, they are considered here as having the same mathematical meaning.</ref>}}
{{defn|no=1|[[Function composition]]: If {{mvar|f}} and {{mvar|g}} are two functions, then <math>g\circ f</math> is the function such that <math>(g\circ f)(x)=g(f(x))</math> for every value of {{mvar|x}}.}}
{{defn|no=2|[[Hadamard product (matrices)|Hadamard product of matrices]]: If {{mvar|A}} and {{mvar|B}} are two matrices of the same size, then <math>A\circ B</math> is the matrix such that <math>(A\circ B)_{i,j} = (A)_{i,j}(B)_{i,j}</math>. Possibly, <math>\circ</math> is also used instead of [[#⊙|{{math|⊙}}]] for the [[Hadamard product (series)|Hadamard product of power series]].{{citation needed|date=November 2020}}}}

{{term|∂|{{math|∂}}}}
{{defn|no=1|[[Boundary (topology)|Boundary]] of a [[topological subspace]]: If {{mvar|S}} is a subspace of a topological space, then its ''boundary'', denoted <math>\partial S</math>, is the [[set difference]] between the [[closure (topology)|closure]] and the [[interior (topology)|interior]] of {{mvar|S}}.}}
{{defn|no=2|[[Partial derivative]]: see [[#partial|{{math|{{sfrac|∂□|∂□}}}}]].}}

{{term|integral|content={{math|∫}}}}
{{defn|no=1|1=Without a subscript, denotes an [[antiderivative]]. For example, <math>\textstyle\int x^2 dx = \frac{x^3}3 +C</math>.}}
{{defn|no=2|1=With a subscript and a superscript, or expressions placed below and above it, denotes a [[definite integral]]. For example, <math>\textstyle \int_a^b x^2dx = \frac{b^3-a^3}{3}</math>.}}
{{defn|no=3|1=With a subscript that denotes a curve, denotes a [[line integral]]. For example, <math>\textstyle\int_C f=\int_a^b f(r(t))r'(t)\operatorname{d}t</math>, if {{mvar|r}} is a parametrization of the curve {{mvar|C}}, from {{mvar|a}} to {{mvar|b}}.}}

{{term|oint|content={{math|∮}}}}
{{defn|Often used, typically in physics, instead of <math>\textstyle\int</math> for [[line integral]]s over a [[closed curve]].}}

{{term|iint|content={{math|∬, ∯}}}}
{{defn|Similar to <math>\textstyle\int</math> and <math>\textstyle\oint</math> for [[surface integral]]s.}}

{{term|∇|[[nabla symbol|<math>\boldsymbol{\nabla}</math> or <math>\vec{\nabla}</math>]]}}
{{defn|[[del|Nabla]], the [[gradient]], vector derivative operator <math>\textstyle \left(\frac \partial {\partial x}, \frac \partial {\partial y}, \frac \partial {\partial z}\right)</math>, also called ''del'' or ''grad'',}} or the [[covariant derivative]].

{{term|Laplacian|{{math|&nabla;<sup>2</sup>}} or {{math|&nabla;&sdot;&nabla;}}}}
{{defn|[[Laplace operator]] or ''Laplacian'': <math>\textstyle \frac {\partial^2}{\partial x^2} + \frac {\partial^2}{\partial y^2} + \frac {\partial^2}{\partial z^2}</math>. The forms <math>\nabla^2</math> and <math>\boldsymbol\nabla \cdot \boldsymbol\nabla</math> represent the dot product of the [[#∇|gradient]] (<math>\boldsymbol{\nabla}</math> or <math>\vec{\nabla}</math>) with itself. Also notated {{math|&Delta;}} (next item).}}

{{term|Delta|{{math|&Delta;}}}}
(Capital Greek letter [[Delta (letter)|delta]]—not to be confused with <math>\triangle</math>, which may denote a geometric [[triangle]] or, alternatively, the [[symmetric difference]] of two sets.)
{{defn|no=1|Another notation for the [[Laplace operator|Laplacian]] (see above).}}
{{defn|no=2|Operator of [[finite difference]].}}

{{term|four-gradient|<math>\boldsymbol{\partial}</math> or <math>\partial_\mu</math>}}
(Note: the notation <math>\Box</math> is not recommended for the four-gradient since both <math>\Box</math> and <math>{\Box}^2</math> are used to denote the [[d'Alembertian]]; see below.)
{{defn|Quad, the 4-vector gradient operator or [[four-gradient]], <math>\textstyle \left( \frac \partial {\partial t}, \frac \partial {\partial x}, \frac \partial {\partial y}, \frac \partial {\partial z}\right)</math>.}}

{{term|d'Alembertian|<math>\Box</math> or <math>{\Box}^2</math>}} (here an actual box, not a placeholder)
{{defn|Denotes the [[d'Alembertian]] or squared [[four-gradient]], which is a generalization of the [[Laplacian]] to four-dimensional spacetime. In flat spacetime with Euclidean coordinates, this may mean either <math>~ \textstyle - \frac {\partial^2}{\partial t^2} + \frac {\partial^2}{\partial x^2} + \frac {\partial^2}{\partial y^2} + \frac {\partial^2}{\partial z^2} ~\;</math>  or  <math>\;~ \textstyle + \frac {\partial^2}{\partial t^2} - \frac {\partial^2}{\partial x^2} - \frac {\partial^2}{\partial y^2} - \frac {\partial^2}{\partial z^2} ~\;</math>; the sign convention must be specified. In curved spacetime (or flat spacetime with non-Euclidean coordinates), the definition is more complicated. Also called ''box'' or ''quabla''.}}

{{glossary end}}

== Linear and multilinear algebra ==
{{glossary}}
{{term|\sum|content={{math|∑}} {{spaces|3|em}}([[capital-sigma notation]])}}
{{defn|no=1|Denotes the [[summation|sum]] of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in <math>\textstyle \sum_{i=1}^n i^2</math> or <math>\textstyle \sum_{0<i<j<n} j-i</math>.}}
{{defn|no=2|Denotes a [[series (mathematics)|series]] and, if the series is [[convergent series|convergent]], the [[sum of series|sum of the series]]. For example, <math>\textstyle \sum_{i=0}^\infty \frac {x^i}{i!}=e^x</math>.}}

{{term|\prod|content={{math|∏}}{{spaces|3|em}} ([[capital-pi notation]])}} 
{{defn|no=1|Denotes the [[product (mathematics)#Product of sequences|product]] of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in <math>\textstyle \prod_{i=1}^n i^2</math> or <math>\textstyle \prod_{0<i<j<n} j-i</math>.}}
{{defn|no=2|Denotes an [[infinite product]]. For example, the [[Riemann zeta function#Euler's product formula|Euler product formula for the Riemann zeta function]] is <math>\textstyle\zeta(z) = \prod_{n=1}^{\infty} \frac{1}{1 - p_n^{-z}}</math>.}}
{{defn|no=3|Also used for the [[Cartesian product]] of any number of sets and the [[direct product]] of any number of [[mathematical structure]]s.}}

{{term|⊕|<math>\oplus</math>}}
{{defn|no=1|Internal [[direct sum]]: if {{mvar|E}} and {{mvar|F}} are abelian subgroups of an [[abelian group]] {{mvar|V}}, notation <math>V=E\oplus F</math> means that {{mvar|V}} is the direct sum of {{mvar|E}} and {{mvar|F}}; that is, every element of {{mvar|V}} can be written in a unique way as the sum of an element of {{mvar|E}} and an element of {{mvar|F}}. This applies also when {{mvar|E}} and {{mvar|F}} are [[linear subspace]]s or [[submodule]]s of the [[vector space]] or [[module (mathematics)|module]] {{mvar|V}}.}}
{{defn|no=2|[[Direct sum]]: if  {{mvar|E}} and {{mvar|F}} are two [[abelian group]]s, [[vector space]]s, or [[module (mathematics)|module]]s, then their direct sum, denoted <math>E\oplus F</math> is an abelian group, vector space, or module (respectively) equipped with two [[monomorphism]]s <math>f:E\to E\oplus F</math> and <math>g:F\to E\oplus F</math> such that <math>E\oplus F</math> is the internal direct sum of <math>f(E)</math> and <math>g(F)</math>. This definition makes sense because this direct sum is unique up to a unique [[isomorphism]].}}
{{defn|no=3|[[Exclusive or]]: if {{mvar|E}} and {{mvar|F}} are two [[Boolean variable]]s or [[predicate (mathematical logic)#Simplified overview|predicate]]s, <math>E\oplus F</math> may denote the exclusive or. Notations {{math|''E'' <small>'''XOR'''</small> ''F''}} and <math>E\veebar F</math> are also commonly used; see [[#⊻|⊻]].}}

{{term|⊗|<math>\otimes</math>}}
{{defn|no=1|Denotes the [[tensor product]] of [[abelian group]]s, [[vector space]]s,  [[module (mathematics)|module]]s, or other mathematical structures, such as in <math>E\otimes F,</math> or <math>E\otimes_K F.</math>}}
{{defn|no=2|Denotes the [[tensor product]] of elements: if <math>x\in E</math> and <math>y\in F,</math> then <math>x\otimes y\in E\otimes F.</math>}}

{{term|□⊤|{{math|□{{sup|⊤}}}}}}
{{defn|no=1|[[Transpose]]: if {{mvar|A}} is a matrix, <math>A^\top</math> denotes the ''transpose'' of {{mvar|A}}, that is, the matrix obtained by exchanging rows and columns of {{mvar|A}}. Notation <math>^\top\!\! A</math> is also used. The symbol <math>\top</math> is often replaced by the letter {{math|T}} or {{mvar|t}}.}}
{{defn|no=2|For inline uses of the symbol, see [[#⊤|⊤]].}}

{{term|□⊥|{{math|□{{sup|⊥}}}}}}
{{defn|no=1|[[Orthogonal complement]]: If {{mvar|W}} is a [[linear subspace]] of an [[inner product space]] {{mvar|V}}, then <math>W^\bot</math> denotes its ''orthogonal complement'', that is, the linear space of the elements of {{mvar|V}} whose inner products with the elements of {{mvar|W}} are all zero.}}
{{defn|no=2|[[Orthogonal subspace]] in the [[dual space]]: If {{mvar|W}} is a [[linear subspace]] (or a [[submodule]]) of a [[vector space]] (or of a [[module (mathematics)|module]]) {{mvar|V}}, then <math>W^\bot</math> may denote the ''orthogonal subspace'' of {{mvar|W}}, that is, the set of all [[linear forms]] that map {{mvar|W}} to zero.}}
{{defn|no=3|For inline uses of the symbol, see [[#⊥|⊥]].}}
{{glossary end}}

== Advanced group theory ==
{{glossary}}
{{term|⊲|{{math|⊲}}{{br}}{{math|⊴}}}}
{{defn|[[Normal subgroup]] of and normal subgroup of including equality, respectively. If {{mvar|N}} and {{mvar|G}} are groups such that {{mvar|N}} is a normal subgroup of (including equality) {{mvar|G}}, this is written <math>N\trianglelefteq G</math>.}}
{{term|⋉|{{math|⋉}}{{br}}{{math|⋊}}}}
{{defn|no=1|Inner [[semidirect product]]: if {{mvar|N}} and {{mvar|H}} are subgroups of a [[group (mathematics)|group]] {{mvar|G}}, such that {{mvar|N}} is a [[normal subgroup]] of {{mvar|G}}, then <math>G=N\rtimes H</math> and <math>G=H\ltimes N</math> mean that {{mvar|G}} is the semidirect product of {{mvar|N}} and {{mvar|H}}, that is, that every element of {{mvar|G}} can be uniquely decomposed as the product of an element of {{mvar|N}} and an element of {{mvar|H}}. (Unlike for the [[direct product of groups]], the element of {{mvar|H}} may change if the order of the factors is changed.)}} 
{{defn|no=2|Outer [[semidirect product]]: if {{mvar|N}} and {{mvar|H}} are two [[group (mathematics)|groups]], and <math>\varphi</math> is a [[group homomorphism]] from {{mvar|N}} to the [[automorphism group]] of {{mvar|H}}, then <math>N\rtimes_\varphi H = H\ltimes_\varphi N</math> denotes a group {{mvar|G}}, unique up to a [[group isomorphism]], which is a semidirect product of {{mvar|N}} and {{mvar|H}}, with the commutation of elements of {{mvar|N}} and {{mvar|H}} defined by <math>\varphi</math>.}}

{{term|≀|{{math|big=1|≀}}}}
{{defn|In [[group theory]], <math>G\wr H</math> denotes the [[wreath product]] of the [[group (mathematics)|groups]] {{math|G}} and {{math|H}}. It is also denoted as <math>G\operatorname{wr} H</math> or <math>G\operatorname{Wr} H</math>; see {{slink|Wreath product|Notation and conventions}} for several notation variants.}}

{{glossary end}}

== Infinite numbers ==
{{glossary}}
{{term|infinity|content=<math>\infty</math> {{spaces|3|em}}([[infinity symbol]])}}
{{defn|no=1|The symbol is read as [[infinity (mathematics)|infinity]]. As an upper bound of a [[summation]], an [[infinite product]], an [[integral]], etc., means that the computation is unlimited. Similarly, <math>-\infty</math> in a lower bound means that the computation is not limited toward negative values.}}
{{defn|no=2|<math>-\infty</math> and <math>+\infty</math> are the generalized numbers that are added to the [[real line]] to form the [[extended real line]].}}
{{defn|no=3|<math>\infty</math> is the generalized number that is added to the real line to form the [[projectively extended real line]].}}

{{term|𝔠|content=<math>\mathfrak c</math>{{spaces|3|em}}([[fraktur]] 𝔠)}}
{{defn|<math>\mathfrak c</math> denotes the [[cardinality of the continuum]], which is the [[cardinality]] of the set of [[real number]]s.}}

{{term|ℵ|<math>\aleph</math>{{spaces|3|em}}([[aleph]])}}
{{defn|With an [[ordinal number|ordinal]] {{mvar|i}} as a subscript, denotes the {{mvar|i}}th [[aleph number]], that is the {{mvar|i}}th infinite [[cardinal number|cardinal]]. For example, <math>\aleph_0</math> is the smallest infinite cardinal, that is, the cardinal of the natural numbers.}}

{{term|ℶ|content=<math>\beth</math>{{spaces|3|em}}([[bet (letter)]])}}
{{defn|With an [[ordinal number|ordinal]] {{mvar|i}} as a subscript, denotes the {{mvar|i}}th [[beth number]]. For example, <math>\beth_0</math> is the [[cardinal number|cardinal]] of the natural numbers, and <math>\beth_1</math> is the [[cardinal of the continuum]].}}

{{term|omega|content=<math>\omega</math>{{spaces|3|em}}([[omega]])}}
{{defn|no=1|Denotes the first [[limit ordinal]]. It is also denoted <math>\omega_0</math> and can be identified with the [[ordered set]] of the [[natural number]]s.}}
{{defn|no=2|With an [[ordinal number|ordinal]] {{mvar|i}} as a subscript, denotes the {{mvar|i}}th [[limit ordinal]] that has a [[cardinality]] greater than that of all preceding ordinals.}}
{{defn|no=3|In [[computer science]], denotes the (unknown) greatest lower bound for the exponent of the [[computational complexity]] of [[Matrix multiplication#Complexity|matrix multiplication]].}}
{{defn|no=4|Written as a [[function (mathematics)|function]] of another function, it is used for comparing the [[asymptotic growth]] of two functions. See {{slink|Big O notation|Related asymptotic notations}}.}}
{{defn|no=5|In [[number theory]], may denote the [[prime omega function]]. That is, <math>\omega(n)</math> is the number of distinct prime factors of the integer {{mvar|n}}.}}

{{glossary end}}

== Brackets ==
Many types of [[bracket (mathematics)|bracket]] are used in mathematics. Their meanings depend not only on their shapes, but also on the nature and the arrangement of what is delimited by them, and sometimes what appears between or before them. For this reason, in the entry titles, the symbol {{math|□}} is used as a placeholder for schematizing the syntax that underlies the meaning.

=== Parentheses ===
{{glossary}}
{{term|()|content={{math|(□)}}}}
{{defn|Used in an [[expression (mathematics)|expression]] for specifying that the sub-expression between the parentheses has to be considered as a single entity; typically used for specifying the [[order of operations]].}}

{{term|functional|content={{math|□(□)}}{{br}}{{math|□(□, □)}}{{br}}{{math|□(□, ..., □)}}}}
{{defn|no=1|[[Functional notation]]: if the first <math>\Box</math> is the name (symbol) of a [[function (mathematics)|function]], denotes the value of the function applied to the expression between the parentheses; for example, <math>f(x)</math>, <math>\sin(x+y)</math>. In the case of a [[multivariate function]], the parentheses contain several expressions separated by commas, such as <math>f(x,y)</math>.}}
{{defn|no=2|May also denote a product, such as in <math>a(b+c)</math>. When the confusion is possible, the context must distinguish which symbols denote functions, and which ones denote [[variable (mathematics)|variables]].}}

{{term|pair|content={{math|(□, □)}}}}
{{defn|no=1|Denotes an [[ordered pair]] of [[mathematical object]]s, for example, <math>(\pi, 0)</math>.}}
{{defn|no=2|If {{mvar|a}} and {{mvar|b}} are [[real number]]s, <math>-\infty</math>, or <math>+\infty</math>, and {{math|''a'' < ''b''}}, then <math>(a,b)</math> denotes the [[open interval]] delimited by {{mvar|a}} and {{mvar|b}}. See [[#open interval|{{math|]□, □[}}]] for an alternative notation.}} 
{{defn|no=3|If {{mvar|a}} and {{mvar|b}} are [[integer]]s, <math>(a,b)</math> may denote the [[greatest common divisor]] of {{mvar|a}} and {{mvar|b}}. Notation <math>\gcd(a,b)</math> is often used instead.}}

{{term|(□,□,□)|content={{math|(□, □, □)}}}}
{{defn|If {{math|''x'', ''y'', ''z''}} are vectors in <math>\mathbb R^3</math>, then <math>(x,y,z)</math> may denote the [[scalar triple product]].{{citation needed|date=November 2020}}  See also [[#sqb3|[□,□,□]]] in {{slink||Square brackets}}.}}

{{term|tuple|content={{math|(□, ..., □)}}}}
{{defn|Denotes a [[tuple]]. If there are {{mvar|n}} objects separated by commas, it is an {{mvar|n}}-tuple.}}

{{term|sequence|content={{math|(□, □, ...)}}{{br}}{{math|(□, ..., □, ...)}}}}
{{defn|Denotes an [[infinite sequence]].}}

{{term|pmatrix|content=<math>\begin{pmatrix}
\Box & \cdots & \Box \\
\vdots & \ddots & \vdots \\
\Box & \cdots & \Box
\end{pmatrix}</math>}}
{{defn|Denotes a [[matrix (mathematics)|matrix]]. Often denoted with [[#bmatrix|square brackets]].}}

{{term|binomial|content=<math>\binom{\Box}{\Box}</math>}}
{{defn|Denotes a [[binomial coefficient]]: Given two [[nonnegative integer]]s, <math>\binom{n}{k}</math> is read as "{{mvar|n}} choose {{mvar|k}}", and is defined as the integer <math>\frac{n(n-1)\cdots(n-k+1)}{1\cdot 2\cdots k}=\frac{n!}{k!\,(n-k)!}</math> (if {{math|''k'' {{=}} 0}}, its value is conventionally {{math|1}}). Using the left-hand-side expression, it denotes a [[polynomial]] in {{mvar|n}}, and is thus defined and used for any [[real number|real]] or [[complex number|complex]] value of {{mvar|n}}.}}

{{term|legendre|content=<math>\left(\frac{\Box}{\Box}\right)</math>}}
{{defn|[[Legendre symbol]]: If {{mvar|p}} is an odd [[prime number]] and {{mvar|a}} is an [[integer]], the value of <math>\left(\frac{a}{p}\right)</math> is 1 if {{mvar|a}} is a [[quadratic residue]] modulo {{mvar|p}}; it is −1 if {{mvar|a}} is a [[quadratic non-residue]] modulo {{mvar|p}}; it is 0 if {{mvar|p}} divides {{mvar|a}}. The same notation is used for the [[Jacobi symbol]] and [[Kronecker symbol]], which are generalizations where {{mvar|p}} is respectively any odd positive integer, or any integer.}}
{{glossary end}}

=== Square brackets ===
{{glossary}}
{{term|sqb1|content={{math|[□]}}}}
{{defn|no=1|Sometimes used as a synonym of [[#()|{{math|(□)}}]] for avoiding nested parentheses.}}
{{defn|no=2|[[Equivalence class]]: given an [[equivalence relation]], <math>[x]</math> often denotes the equivalence class of the element {{mvar|x}}.}}
{{defn|no=3|[[Integral part]]: if {{mvar|x}} is a [[real number]], <math>[x]</math> often denotes the integral part or [[truncation]] of {{mvar|x}}, that is, the integer obtained by removing all digits after the [[decimal mark]]. This notation has also been used for other variants of [[floor and ceiling functions]].}}
{{defn|no=4|[[Iverson bracket]]: if {{mvar|P}} is a [[predicate (mathematical logic)|predicate]], <math>[P]</math> may denote the Iverson bracket, that is the [[function (mathematics)|function]] that takes the value {{math|1}} for the values of the [[free variable]]s in {{mvar|P}} for which {{mvar|P}} is true, and takes the value {{math|0}} otherwise. For example, <math>[x=y]</math> is the [[Kronecker delta function]], which equals one if <math>x=y</math>, and zero otherwise.}}
{{defn|no=5|In combinatorics or computer science, sometimes <math>[n]</math> with <math>n\in\mathbb{N}</math> denotes the set <math>\{1,2,3,\ldots,n\}</math> of positive integers up to {{mvar|n}}, with <math>[0]=\empty</math>.}}

{{term|sqbf|content={{math|□[□]}}}}
{{defn|[[image (mathematics)|Image of a subset]]: if {{mvar|S}} is a [[subset]] of the [[domain of a function|domain of the function]] {{mvar|f}}, then <math>f[S]</math> is sometimes used for denoting the image of {{mvar|S}}. When no confusion is possible, notation [[#functional|{{math|''f''(''S'')}}]] is commonly used.}}

{{term|sqb2|content={{math|[□, □]}}}}
{{defn|no=1|[[Closed interval]]: if {{mvar|a}} and {{mvar|b}} are [[real number]]s such that <math>a\le b</math>, then <math>[a,b]</math> denotes the closed interval defined by them.}}
{{defn|no=2|[[Commutator (group theory)]]: if {{mvar|a}} and {{mvar|b}} belong to a [[group (mathematics)|group]], then <math>[a,b]=a^{-1}b^{-1}ab</math>.}}
{{defn|no=3|[[Commutator (ring theory)]]: if {{mvar|a}} and {{mvar|b}} belong to a [[ring (mathematics)|ring]], then <math>[a,b]=ab-ba</math>.}}
{{defn|no=4|Denotes the [[Lie bracket]], the operation of a [[Lie algebra]].}}

{{term|sqb2:|content={{math|[□ : □]}}}}
{{defn|no=1|[[Degree of a field extension]]: if {{mvar|F}} is an [[extension field|extension]] of a [[field (mathematics)|field]] {{mvar|E}}, then <math>[F:E]</math> denotes the degree of the [[field extension]] <math>F/E</math>. For example, <math>[\mathbb C:\mathbb R]=2</math>.}}
{{defn|no=2|[[Index of a subgroup]]: if {{mvar|H}} is a [[subgroup]] of a [[group (mathematics)|group]] {{mvar|E}}, then <math>[G:H]</math> denotes the index of {{mvar|H}} in {{mvar|G}}. The notation [[#!:!|{{math|{{abs|G:H}}}}]] is also used}}

{{term|sqb3|content={{math|[□, □, □]}}}}
{{defn|If {{math|''x'', ''y'', ''z''}} are vectors in <math>\mathbb R^3</math>, then <math>[x,y,z]</math> may denote the [[scalar triple product]].<ref>{{cite book|last = Rutherford |first = D. E. |title = Vector Methods |publisher = Oliver and Boyd Ltd., Edinburgh |series = University Mathematical Texts |year = 1965}}</ref>  See also [[#(□,□,□)|(□,□,□)]] in {{slink||Parentheses}}.}}

{{term|bmatrix|content=<math>\begin{bmatrix}
\Box & \cdots & \Box \\
\vdots & \ddots & \vdots \\
\Box & \cdots & \Box
\end{bmatrix}</math>}}
{{defn|Denotes a [[matrix (mathematics)|matrix]]. Often denoted with [[#pmatrix|parentheses]].}}

{{glossary end}}

=== Braces ===
{{glossary}}
{{term|bb|content={{math|{{mset| }}}}}}
{{defn|[[Set-builder notation]] for the [[empty set]], also denoted <math>\emptyset</math> or [[#∅|∅]].}}

{{term|b□b|content={{math|{{brace|□}} }}}}
{{defn|no=1|Sometimes used as a synonym of [[#()|{{math|(□)}}]] and [[#sqb1|{{math|[□]}}]] for avoiding nested parentheses.}}
{{defn|no=2|[[Set-builder notation]] for a [[singleton set]]: <math>\{x\}</math> denotes the [[set (mathematics)|set]] that has {{mvar|x}} as a single element.}}

{{term|b,...,b|content={{math|{{brace|□, ..., □}} }}}}
{{defn|[[Set-builder notation]]: denotes the [[set (mathematics)|set]] whose elements are listed between the braces, separated by commas.}}

{{term|b:b|content={{math|{{brace|□ : □}} }}{{br}}{{math|{{brace|□ {{!}} □}} }}}}
{{defn|[[Set-builder notation]]: if <math>P(x)</math> is a [[predicate (mathematical logic)|predicate]] depending on a [[variable (mathematics)|variable]] {{mvar|x}}, then both <math>\{x : P(x)\}</math> and <math>\{x\mid P(x)\}</math> denote the [[set (mathematics)|set]] formed by the values of {{mvar|x}} for which <math>P(x)</math> is true.}}

{{term|Single brace}}
{{defn|no=1|Used for emphasizing that several [[equation (mathematics)|equations]] have to be considered as [[simultaneous equations]]; for example, <math>\textstyle \begin{cases}2x+y=1\\3x-y=1\end{cases}</math>.}}
{{defn|no=2|[[Piecewise]] definition; for example, <math>\textstyle |x|=\begin{cases}x&\text{if }x\ge 0\\-x&\text{if }x< 0\end{cases}</math>.}}
{{defn|no=3|Used for grouped annotation of elements in a formula; for example, <math>\textstyle \underbrace{ (a,b,\ldots,z) }_{26}</math>, <math>\textstyle \overbrace{ 1+2+\cdots+100 }^{=5050}</math>, <math>\textstyle \left.\begin{bmatrix}A\\B\end{bmatrix}\right\} m+n\text{ rows}</math>}}

{{glossary end}}

=== Other brackets ===

{{glossary}}
{{term|!□!|content={{math|{{!}}□{{!}}}}}}
{{defn|no=1|[[Absolute value]]: if {{mvar|x}} is a [[real number|real]] or [[complex number|complex]] number, <math>|x|</math> denotes its absolute value.}}
{{defn|no=2|Number of elements: If {{math|S}} is a [[set (mathematics)|set]], <math>|S|</math> may denote its [[cardinality]], that is, its number of elements. <math>\#S</math> is also often used, see [[#sharp|{{math|#}}]].}}
{{defn|no=3|Length of a [[line segment]]: If {{mvar|P}} and {{mvar|Q}} are two points in a [[Euclidean space]], then <math>|PQ|</math> often denotes the length of the line segment that they define, which is the [[Euclidean norm|distance]] from {{mvar|P}} to {{mvar|Q}}, and is often denoted <math>d(P,Q)</math>.}}
{{defn|no=4|For a similar-looking operator, see [[#vbar|{{math|{{!}}}}]].}}

{{term|!:!|content={{math|{{abs|□:□}}}}}}
{{defn|[[Index of a subgroup]]: if {{mvar|H}} is a [[subgroup]] of a [[group (mathematics)|group]] {{mvar|G}}, then <math>|G:H|</math> denotes the index of {{mvar|H}} in {{mvar|G}}. The notation [[#sqb2:|{{math|[G:H]}}]] is also used}}

{{term|determinant|content=<math>\textstyle\begin{vmatrix}
\Box & \cdots & \Box \\
\vdots & \ddots & \vdots \\
\Box & \cdots & \Box
\end{vmatrix}</math>}}
{{defn|<math>\begin{vmatrix}
x_{1,1} & \cdots & x_{1,n} \\
\vdots & \ddots & \vdots \\
x_{n,1} & \cdots & x_{n,n}
\end{vmatrix}</math> denotes the [[determinant]] of the [[square matrix]] <math>\begin{bmatrix}
x_{1,1} & \cdots & x_{1,n} \\
\vdots & \ddots & \vdots \\
x_{n,1} & \cdots & x_{n,n}
\end{bmatrix}</math>.}}

{{term|norm|content={{math|{{!!}}□{{!!}}}}}}
{{defn|no=1|Denotes the [[norm (mathematics)|norm]] of an element of a [[normed vector space]].}}
{{defn|no=2|For the similar-looking operator named ''parallel'', see [[#∥|{{math|∥}}]].}}

{{term|⌊⌋|content={{math|⌊□⌋}}}}
{{defn|[[Floor function]]: if {{mvar|x}} is a real number, <math>\lfloor x\rfloor</math> is the greatest [[integer]] that is not greater than {{mvar|x}}.}}

{{term|⌈⌉|content={{math|⌈□⌉}}}}
{{defn|[[Ceiling function]]: if {{mvar|x}} is a real number, <math>\lceil x\rceil</math> is the lowest [[integer]] that is not lesser than {{mvar|x}}.}}

{{term|⌊⌉|content= {{math|⌊□⌉}}}}
{{defn|[[Nearest integer function]]: if {{mvar|x}} is a real number, <math>\lfloor x\rceil</math> is the  [[integer]] that is the closest to {{mvar|x}}.}}

{{term|open interval|content={{math|]□, □[}}}}
{{defn|[[Open interval]]: If a and b are real numbers, <math>-\infty</math>, or <math>+\infty</math>, and <math>a<b</math>, then 
<math>]a,b[</math> denotes the open interval delimited by a and b. See [[#pair|{{math|(□, □)}}]] for an alternative notation.}}

{{term|left-open|content={{math|(□, □]}}{{br}}{{math|]□, □]}}}}
{{defn|Both notations are used for a [[half-open interval|left-open interval]].}}

{{term|right-open|content={{math|[□, □)}}{{br}}{{math|[□, □[}}}}
{{defn|Both notations are used for a [[half-open interval|right-open interval]].}}

{{term|⟨⟩|content={{math|⟨□⟩}}}}
{{defn|no=1|[[Generating set|Generated object]]: if {{math|S}} is a set of elements in an algebraic structure, <math>\langle S \rangle</math> denotes often the object generated by {{math|S}}. If <math>S=\{s_1,\ldots, s_n\}</math>, one writes <math>\langle s_1,\ldots, s_n \rangle</math> (that is, braces are omitted). In particular, this may denote
* the [[linear span]] in a [[vector space]] (also often denoted {{math|Span(''S'')}}),
* the generated [[subgroup]] in a [[group (mathematics)|group]],
* the generated [[ideal (ring theory)|ideal]] in a [[ring (mathematics)|ring]],
* the generated [[submodule]] in a [[module (mathematics)|module]].}}
{{defn|no=2|Often used, mainly in physics, for denoting an [[expected value]]. In [[probability theory]], <math>E(X)</math> is generally used instead of <math>\langle S \rangle</math>.}}
{{term|⟨,⟩|content={{math|⟨□, □⟩}}{{br}}{{math|⟨□ {{!}} □⟩}}}}
{{defn|Both <math>\langle x, y\rangle</math> and <math>\langle x\mid y\rangle</math> are commonly used for denoting the [[inner product]] in an [[inner product space]].}}

{{term|bra–ket|content=<math>\langle\Box| \text{ and } |\Box\rangle</math>}}
{{defn|[[Bra–ket notation]] or ''Dirac notation'': if {{mvar|x}} and {{mvar|y}} are elements of an [[inner product space]], <math>|x\rangle</math> is the vector defined by {{mvar|x}}, and <math>\langle y|</math> is the [[covector]] defined by {{mvar|y}}; their inner product is <math>\langle y\mid x\rangle</math>.}}
{{glossary end}}

== Symbols that do not belong to formulas ==
In this section, the symbols that are listed are used as some sorts of punctuation marks in mathematical reasoning, or as abbreviations of natural language phrases. They are generally not used inside a formula. Some were used in [[classical logic]] for indicating the logical dependence between sentences written in plain language. Except for the first two, they are normally not used in printed mathematical texts since, for readability, it is generally recommended to have at least one word between two formulas. However, they are still used on a [[black board]] for indicating relationships between formulas.

{{glossary}}
{{term|qed|content=[[tombstone (typography)|{{math|■ , □}}]]}}
{{defn|Used for marking the end of a proof and separating it from the current text. The [[initialism]] [[Q.E.D.|Q.E.D. or QED]] ({{langx|la|quod erat demonstrandum}}, "as was to be shown") is often used for the same purpose, either in its upper-case form or in lower case.}}

{{term|☡|[[Bourbaki dangerous bend symbol|{{math|size=200%|☡}}]]}}
{{defn|[[Bourbaki dangerous bend symbol]]: Sometimes used in the margin to forewarn readers against serious errors, where they risk falling, or to mark a passage that is tricky on a first reading because of an especially subtle argument.}}

{{term|therefore|content=[[therefore sign|{{math|∴}}]]}}
{{defn|Abbreviation of "therefore". Placed between two assertions, it means that the first one implies the second one. For example: "All humans are mortal, and Socrates is a human. ∴ Socrates is mortal."}}

{{term|because|content=[[because sign|{{math|∵}}]]}}
{{defn|Abbreviation of "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: "{{math|11}} is [[prime number|prime]] ∵ it has no positive integer factors other than itself and one."}}

{{term|∋|content={{math|∋}}}}
{{defn|no=1|Abbreviation of "such that". For example, <math>x\ni x>3</math> is normally printed "{{mvar|x}} such that <math>x>3</math>".}}
{{defn|no=2|Sometimes used for reversing the operands of <math>\in</math>; that is, <math>S\ni x</math> has the same meaning as <math>x\in S</math>. See [[#∈|∈]] in {{slink||Set theory}}.}}

{{term|∝|content={{math|∝}}}}
{{defn|Abbreviation of "is proportional to".}}

{{glossary end}}

== Miscellaneous ==
{{glossary}}
{{term|!|content=[[!|{{math|!}}]]}}
{{defn|no=1|[[Factorial]]: if {{mvar|n}} is a [[positive integer]], {{math|''n''!}} is the product of the first {{mvar|n}} positive integers, and is read as "n factorial".}}
{{defn|no=2|[[Double factorial]]: if {{mvar|n}} is a [[positive integer]], {{math|''n''!!}} is the product of all positive integers up to {{mvar|n}} with the same parity as {{mvar|n}}; that is, if {{mvar|n}} is odd, the product of all odd integers from 1 up to and including {{mvar|n}}, and if {{mvar|n}} is even, the product of all even integers, up to and including  {{mvar|n}}. It is read as "the double factorial of {{mvar|n}}".}}
{{defn|no=3|[[Subfactorial]]: if {{mvar|n}} is a positive integer, {{math|!''n''}} is the number of [[derangements]] of a set of {{mvar|n}} elements, and is read as "the subfactorial of n".}}

{{term|*|content=[[*|{{math|*}}]]}}
{{defn|Many different uses in mathematics; see ''{{slink|Asterisk|Mathematics}}''.}}

{{term|vbar|content=[[vertical bar|{{math|{{!}}}}]]}}
{{defn|no=1|[[Divisibility]]: if {{mvar|m}} and {{mvar|n}} are two integers, <math>m\mid n</math> means that {{mvar|m}} divides {{mvar|n}} evenly.}}
{{defn|no=2|In [[set-builder notation]], it is used as a separator meaning "such that"; see [[#b:b|{{math|{{mset|□ {{!}} □}}}}]].}}
{{defn|no=3|[[restriction (mathematics)|Restriction of a function]]: if {{mvar|f}} is a [[function (mathematics)|function]], and {{mvar|S}} is a [[subset]] of its [[domain of a function|domain]], then <math>f|_S</math> is the function with {{mvar|S}} as a domain that equals {{mvar|f}} on {{mvar|S}}.}}
{{defn|no=4|[[Conditional probability]]: <math>P(X\mid E)</math> denotes the probability of {{mvar|X}} given that the event {{mvar|E}} occurs. Also denoted <math>P(X/ E)</math>; see "[[#/|/]]".}}
{{defn|no=5|For several uses as [[brackets]] (in pairs or with {{math|⟨}} and {{math|⟩}}) see ''{{slink||Other brackets}}''.}}

{{term|∤|content={{math|∤}}}}
{{defn|[[Divisibility|Non-divisibility]]: <math>n\nmid m</math> means that {{mvar|n}} is not a divisor of {{mvar|m}}.}}

{{term|∥|content =[[∥|{{math|∥}}]]}}
{{defn|no=1|Denotes [[parallel (geometry)|parallelism]] in [[elementary geometry]]: if {{mvar|PQ}} and {{mvar|RS}} are two [[line (geometry)|lines]], <math>PQ\parallel RS</math> means that they are parallel.}}
{{defn|no=2|[[Parallel (operator)|Parallel]] – the harmonic sum – an [[operation (mathematics)|arithmetical operation]] used in [[electrical engineering]] for summing two [[electrical impedance|impedances]] wired [[Series and parallel circuits|in parallel]] (e.g. [[parallel resistors]]) or two [[admittance]]s wired [[series wiring|in series]] (e.g. [[Capacitor#series_capacitor_formula_anchor|series capacitors]]): <math>\ x \parallel y = \frac{ 1 }{\ \frac{\ 1\ }{ x } +\frac{\ 1\ }{ y }\ } = \frac{x\ y}{\ x + y\ } ~.</math>}}
{{defn|no=3|Used in pairs as brackets, denotes a [[norm (mathematics)|norm]]; see [[#norm|{{math|{{!!}}□{{!!}}}}]].}}
{{defn|no=4|[[Concatenation]]: Typically used in computer science, <math>x\mathbin{\vert\vert}y</math> is said to represent the value resulting from appending the digits of {{mvar|y}} to the end of {{mvar|x}}.}}
{{defn|no=5|<math>{\displaystyle D_{\text{KL}}(P\parallel Q)}</math>, denotes a [[statistical distance]] or measure of how one [[probability distribution]] P is different from a second, reference probability distribution Q.}}

{{term|∦|content =[[∥|{{math|∦}}]]}}
{{defn|Sometimes used for denoting that two [[line (geometry)|lines]] are not parallel; for example, <math>PQ\not\parallel RS</math>.}}

{{term|⟂|content =[[⟂|<math>\perp</math>]]}}
{{defn|no=1|Denotes [[perpendicularity]] and [[orthogonality]]. For example, if {{mvar|A, B, C}} are three points in a [[Euclidean space]], then <math>AB\perp AC</math> means that the [[line segment]]s {{mvar|AB}} and {{mvar|AC}} are [[perpendicular]], and form a [[right angle]].}}
{{defn|no=2|For the similar symbol, see [[#⊥|<math>\bot</math>]].}}

{{term|⊙|{{math|⊙}}}}
{{defn|[[Hadamard product (series)|Hadamard product of power series]]: if <math>\textstyle S=\sum_{i=0}^\infty s_ix^i</math> and <math>\textstyle T=\sum_{i=0}^\infty t_ix^i</math>, then <math>\textstyle S\odot T=\sum_{i=0}^\infty s_i t_i x^i</math>. Possibly, <math>\odot</math> is also used instead of [[#∘|{{math|○}}]] for the [[Hadamard product (matrices)|Hadamard product of matrices]].{{citation needed|date=November 2020}}}}
{{glossary end}}

== See also ==
=== Related articles ===
* [[Language of mathematics]]
* [[Mathematical notation]]
* [[Notation in probability and statistics]]
* [[Physical constant]]s

=== Related lists ===
* [[List of logic symbols]]
* [[List of mathematical constants]]
* [[Table of mathematical symbols by introduction date]]
* [[Blackboard bold]]
* [[Greek letters used in mathematics, science, and engineering]]
* [[Latin letters used in mathematics, science, and engineering]]
* [[List of common physics notations]]
* [[List of letters used in mathematics, science, and engineering]]
* [[List of mathematical abbreviations]]
* [[List of typographical symbols and punctuation marks]]
* [[ISO 31-11]] (Mathematical signs and symbols for use in physical sciences and technology)
* [[APL syntax and symbols#Monadic functions|List of APL functions]]

=== [[Unicode]] symbols ===
* [[Unicode block]]
* [[Mathematical Alphanumeric Symbols|Mathematical Alphanumeric Symbols (Unicode block)]]
* [[List of Unicode characters]]
* [[Letterlike Symbols]]
* [[Mathematical operators and symbols in Unicode]] 
* Miscellaneous Mathematical Symbols: [[Miscellaneous Mathematical Symbols-A|A]], [[Miscellaneous Mathematical Symbols-B|B]], [[Miscellaneous Technical|Technical]]
* [[Arrow (symbol)]] and [[Miscellaneous Symbols and Arrows]]
* [[Number Forms]]
* [[Geometric Shapes (Unicode block)|Geometric Shapes]]
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== References ==
{{reflist}}

== External links ==
* [http://jeff560.tripod.com/mathsym.html Jeff Miller: ''Earliest Uses of Various Mathematical Symbols'']
* [http://www.numericana.com/answer/symbol.htm Numericana: ''Scientific Symbols and Icons'']
* [http://us.metamath.org/symbols/symbols.html GIF and PNG Images for Math Symbols]
* [https://web.archive.org/web/20070117015443/http://tlt.psu.edu/suggestions/international/bylanguage/math.html Mathematical Symbols in Unicode]
* [https://detexify.kirelabs.org/classify.html Detexify: LaTeX Handwriting Recognition Tool]

; Some Unicode charts of mathematical operators and symbols<nowiki>:</nowiki>
* [https://www.unicode.org/charts/#symbols Index of Unicode symbols]
* [https://www.unicode.org/charts/PDF/U2100.pdf Range 2100–214F: Unicode Letterlike Symbols]
* [https://www.unicode.org/charts/PDF/U2190.pdf Range 2190–21FF: Unicode Arrows]
* [https://www.unicode.org/charts/PDF/U2200.pdf Range 2200–22FF: Unicode Mathematical Operators]
* [https://www.unicode.org/charts/PDF/U27C0.pdf Range 27C0–27EF: Unicode Miscellaneous Mathematical Symbols–A]
* [https://www.unicode.org/charts/PDF/U2980.pdf Range 2980–29FF: Unicode Miscellaneous Mathematical Symbols–B]
* [https://www.unicode.org/charts/PDF/U2A00.pdf Range 2A00–2AFF: Unicode Supplementary Mathematical Operators]

; Some Unicode cross-references<nowiki>:</nowiki>
* [https://web.archive.org/web/20141105143723/http://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Symbols Short list of commonly used LaTeX symbols] and [https://web.archive.org/web/20090323063515/http://mirrors.med.harvard.edu/ctan/info/symbols/comprehensive/ Comprehensive LaTeX Symbol List]
* [https://web.archive.org/web/20140222144828/http://www.robinlionheart.com/stds/html4/entities-mathml MathML Characters] - sorts out Unicode, HTML and MathML/TeX names on one page
* [http://www.w3.org/TR/REC-MathML/chap6/bycodes.html Unicode values and MathML names]
* [https://web.archive.org/web/20141126074509/http://svn.ghostscript.com/ghostscript/branches/gs-db/Resource/Decoding/Unicode Unicode values and Postscript names] from the source code for [[Ghostscript]]

{{Areas of mathematics}}
{{Mathematical symbols notation language}}

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[[Category:Mathematical symbols| ]]
[[Category:Mathematical notation|*]]
[[Category:Lists of symbols|Mathematics]]
[[Category:Mathematical logic|Symbols]]
[[Category:Mathematical tables|Symbols]]
[[Category:Glossaries of mathematics|Symbols]]
[[Category:Wikipedia glossaries using description lists]]