Glossary of mathematical symbols

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

The most basic symbols are thedecimal digits(0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of theLatin alphabet.The decimal digits are used for representing numbers through theHindu–Arabic numeral system.Historically, upper-case letters were used for representingpointsin geometry, and lower-case letters were used forvariablesandconstants.Letters are used for representing many other sorts ofmathematical objects.As the number of these sorts has remarkably increased in modern mathematics, theGreek alphabetand someHebrew lettersare also used. In mathematicalformulas,the standardtypefaceisitalic typefor Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. For having more symbols, other typefaces are also used, mainlyboldface $\mathbf {a,A,b,B},\ldots$ ,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 ${\mathfrak {a,A,b,B}},\ldots$ ,andblackboard 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, seeVariable (mathematics)andList 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 $\textstyle \sum {}$ .

These letters alone are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin inpunctuation marksanddiacriticstraditionally used intypography;others by deformingletter forms,as in the cases of $\in$ and $\forall$ .Others, such as $+$ and $=$ ,were specially designed for mathematics.

Layout of this article

Normally, entries of aglossaryare 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 longsection on bracketshas 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 theirsyntax,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 $\Box$ is used for representing the neighboring parts of a formula that contains the symbol. See§ Bracketsfor examples of use.
Most symbols have two printed versions. They can be displayed asUnicodecharacters, or inLaTeXformat. With the Unicode version, usingsearch enginesandcopy-pastingare 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 ananchor,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

$+$ (plus sign): 1. Denotesadditionand is read asplus;for example, $3 + 2$ .; 2. Denotes that a number ispositiveand is read asplus.Redundant, but sometimes used for emphasizing that a number ispositive,specially when other numbers in the context are or may be negative; for example, $+2$ .; 3. Sometimes used instead of $\sqcup$ for adisjoint unionofsets.
$-$ (minus sign): 1. Denotessubtractionand is read asminus;for example, $3 - 2$ .; 2. Denotes theadditive inverseand is read asminus,the negative of,orthe opposite of;for example, $-2$ .; 3. Also used in place of $\$ for denoting theset-theoretic complement;see\in§ Set theory.
$\times$ (multiplication sign): 1. Inelementary arithmetic,denotesmultiplication,and is read astimes;for example, $3 \times 2$ .; 2. Ingeometryandlinear algebra,denotes thecross product.; 3. Inset theoryandcategory theory,denotes theCartesian productand thedirect product.See also×in§ Set theory.
$\cdot$ (dot): 1. Denotesmultiplicationand is read astimes;for example, $3 \cdot 2$ .; 2. Ingeometryandlinear algebra,denotes thedot product.; 3. Placeholder used for replacing an indeterminate element. For example, saying "theabsolute valueis denoted by $| \cdot |$ "is perhaps clearer than saying that it is denoted as $| |$ .
$\pm$ (plus–minus sign): 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 an unknown value that lies between 8 and 12.
$\mp$ (minus-plus sign): Used paired with $\pm$ ,denotes the opposite sign; that is, $+$ if $\pm$ is $-$ ,and $-$ if $\pm$ is $+$ .
$\div$ (division sign): Widely used for denotingdivisioninAnglophonecountries, it is no longer in common use in mathematics and its use is "not recommended".^[1]In some countries, it can indicate subtraction.
$:$ (colon): 1. Denotes theratioof two quantities.; 2. In some countries, may denotedivision.; 3. Inset-builder notation,it is used as a separator meaning "such that"; see ${□: □}$ .
$/$ (slash): 1. Denotesdivisionand is read asdivided byorover.Often replaced by a horizontal bar. For example, $3 / 2$ or ${\frac {3}{2}}$ .; 2. Denotes aquotient structure.For example,quotient set,quotient group,quotient category,etc.; 3. Innumber theoryandfield theory, $F/E$ denotes afield extension,where $F$ is anextension fieldof thefield $E$ .; 4. Inprobability theory,denotes aconditional probability.For example, $P(A/B)$ denotes the probability of $A$ ,given that $B$ occurs. Usually denoted $P(A\mid B)$ :see " $|$ ".
$\sqrt$ (square-root symbol): Denotessquare rootand is read asthe square root of.Rarely used in modern mathematics without a horizontal bar delimiting the width of its argument (see the next item). For example, $\sqrt2$ .
$\sqrt$ (radical symbol): 1. Denotessquare rootand is read asthe 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, ${\sqrt[{7}]{3}}$ denotes the 7th root of 3.
$^$ (caret): 1.Exponentiationis normally denoted with asuperscript.However, $x^{y}$ is often denoted $x^y$ when superscripts are not easily available, such as inprogramming languages(includingLaTeX) or plain textemails.; 2. Not to be confused with∧

Equality, equivalence and similarity

$=$ (equals sign): 1. Denotesequality.; 2. Used for naming amathematical objectin a sentence like "let $x=E$ ",where $E$ is anexpression.See also $≝$ , $≜$ or ${\displaystyle:=}$ .
$\triangleq \quad {\stackrel {\scriptscriptstyle \mathrm {def} }{=}}\quad:=$: Any of these is sometimes used for naming amathematical object.Thus, $x\triangleq E,$ $x\mathrel {\stackrel {\scriptscriptstyle \mathrm {def} }{=}} E,$ $x\mathrel {:=} E$ and $E\mathrel {=:} x$ are each an abbreviation of the phrase "let $x=E$ ",where⁠ $E$ ⁠is anexpressionand⁠ $x$ ⁠is avariable. This is similar to the concept ofassignmentin computer science, which is variously denoted (depending on theprogramming languageused) $=,:=,\leftarrow,\ldots$
$\neq$ (not-equal sign): Denotesinequalityand means "not equal".
$\approx$: The most common symbol for denotingapproximate equality.For example, $\pi \approx 3.14159.$
$~$ (tilde): 1. Between two numbers, either it is used instead of $\approx$ to mean "approximatively equal", or it means "has the sameorder of magnitudeas ".; 2. Denotes theasymptotic equivalenceof two functions or sequences.; 3. Often used for denoting other types of similarity, for example,matrix similarityorsimilarity of geometric shapes.; 4. Standard notation for anequivalence relation.; 5. Inprobabilityandstatistics,may specify theprobability distributionof arandom variable.For example, $X\sim N(0,1)$ means that the distribution of the random variable $X$ isstandard normal.^[2]; 6. Notation forproportionality.See also $\propto$ for a less ambiguous symbol.
$\equiv$ (triple bar): 1. Denotes anidentity;that is, an equality that is true whichever values are given to the variables occurring in it.; 2. Innumber theory,and more specifically inmodular arithmetic,denotes thecongruencemodulo an integer.; 3. May denote alogical equivalence.
$\cong$: 1. May denote anisomorphismbetween twomathematical structures,and is read as "is isomorphic to".; 2. Ingeometry,may denote thecongruenceof twogeometric shapes(that is the equalityup toadisplacement), and is read "is congruent to".

Comparison

$<$ (less-than sign): 1.Strict inequalitybetween two numbers; means and is read as "less than".; 2. Commonly used for denoting anystrict order.; 3. Between twogroups,may mean that the first one is aproper subgroupof the second one.
$>$ (greater-than sign): 1.Strict inequalitybetween two numbers; means and is read as "greater than".; 2. Commonly used for denoting anystrict order.; 3. Between twogroups,may mean that the second one is aproper subgroupof 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 twogroups,may mean that the first one is asubgroupof 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 twogroups,may mean that the second one is asubgroupof the first one.
$\ll {\text{ and }}\gg$: 1. Means "much less than"and"much greater than".Generally,muchis 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 severalorders of magnitude.; 2. Inmeasure theory, $\mu \ll \nu$ means that the measure $\mu$ is absolutely continuous with respect to the measure $\nu$ .
$\leqq$: A rarely used symbol, generally a synonym of $\leq$ .
$\prec {\text{ and }}\succ$: 1. Often used for denoting anorderor, more generally, apreorder,when it would be confusing or not convenient to use $<$ and $>$ .; 2.Sequentioninasynchronous logic.

Set theory

$\emptyset$: Denotes theempty set,and is more often written $\emptyset$ .Usingset-builder notation,it may also be denoted $\{\}$ .
$#$ (number sign): 1. Number of elements: $\#{}S$ may denote thecardinalityof theset $S$ .An alternative notation is $|S|$ ;see $|\square |$ .; 2.Primorial: $n{}\#$ denotes the product of theprime numbersthat are not greater than $n$ .; 3. Intopology, $M\#N$ denotes theconnected sumof twomanifoldsor twoknots.
$\in$: Denotesset membership,and is read "is in", "belongs to", or "is a member of". That is, $x\in S$ means that $x$ is an element of the set $S$ .
$\notin$: Means "is not in". That is, $x\notin S$ means $\neg (x\in S)$ .
$\subset$: Denotesset inclusion.However two slightly different definitions are common.; 1. $A\subset B$ may mean that $A$ is asubsetof $B$ ,and is possibly equal to $B$ ;that is, every element of $A$ belongs to $B$ ;expressed as a formula, $\forall {}x,\,x\in A\Rightarrow x\in B$ .; 2. $A\subset B$ may mean that $A$ is aproper subsetof $B$ ,that is the two sets are different, and every element of $A$ belongs to $B$ ;expressed as a formula, $A\neq B\land \forall {}x,\,x\in A\Rightarrow x\in B$ .
$\subseteq$: $A\subseteq B$ means that $A$ is asubsetof $B$ .Used for emphasizing that equality is possible, or when $A\subset B$ means that $A$ is a proper subset of $B.$
$⊊$: $A\subsetneq B$ means that $A$ is aproper subsetof $B$ .Used for emphasizing that $A\neq B$ ,or when $A\subset B$ does not imply that $A$ is a proper subset of $B.$
$\supset, \supseteq, ⊋$: Denote the converse relation of $\subset$ , $\subseteq$ ,and $\subsetneq$ respectively. For example, $B\supset A$ is equivalent to $A\subset B$ .
$\cup$: Denotesset-theoretic union,that is, $A\cup B$ is the set formed by the elements of $A$ and $B$ together. That is, $A\cup B=\{x\mid (x\in A)\lor (x\in B)\}$ .
$\cap$: Denotesset-theoretic intersection,that is, $A\cap B$ is the set formed by the elements of both $A$ and $B$ .That is, $A\cap B=\{x\mid (x\in A)\land (x\in B)\}$ .
$∖$ (backslash): 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.
$⊖$ or $\triangle$: Symmetric difference:that is, $A\ominus B$ or $A\operatorname {\triangle } B$ is the set formed by the elements that belong to exactly one of the two sets $A$ and $B$ .
$\complement$: 1. With a subscript, denotes aset complement:that is, if $B\subseteq A$ ,then $\complement _{A}B=A\setminus B$ .; 2. Without a subscript, denotes theabsolute 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 theuniverse of discourse.
$\times$ (multiplication sign): See also×in§ Arithmetic operators.; 1. Denotes theCartesian productof two sets. That is, $A\times B$ is the set formed by allpairsof an element of $A$ and an element of $B$ .; 2. Denotes thedirect productof twomathematical structuresof the same type, which is theCartesian productof the underlying sets, equipped with a structure of the same type. For example,direct product of rings,direct product of topological spaces.; 3. Incategory theory,denotes thedirect product(often called simplyproduct) of two objects, which is a generalization of the preceding concepts of product.
$\sqcup$: Denotes thedisjoint union.That is, if $A$ and $B$ are sets then $A\sqcup B=\left(A\times \{i_{A}\}\right)\cup \left(B\times \{i_{B}\}\right)$ is a set ofpairswhere $i A$ and $i B$ are distinct indices discriminating the members of $A$ and $B$ in⁠ $A\sqcup B$ ⁠.
$\bigsqcup {\text{ or }}\coprod$: 1. Used for thedisjoint unionof a family of sets, such as in ${\textstyle \bigsqcup _{i\in I}A_{i}.}$; 2. Denotes thecoproductofmathematical structuresor of objects in acategory.

Basic logic

Severallogical symbolsare widely used in all mathematics, and are listed here. For symbols that are used only inmathematical logic,or are rarely used, seeList of logic symbols.

$\neg$ (not sign): Denoteslogical negation,and is read as "not". If $E$ is alogical predicate, $\neg E$ is the predicate that evaluates totrueif and only if $E$ evaluates tofalse.For clarity, it is often replaced by the word "not". Inprogramming languagesand some mathematical texts, it is sometimes replaced by " $~$ "or" $!$ ",which are easier to type on some keyboards.
$\lor$ (descending wedge): 1. Denotes thelogical or,and is read as "or". If $E$ and $F$ arelogical predicates, $E\lor F$ is true if either $E$ , $F$ ,or both are true. It is often replaced by the word "or".; 2. Inlattice theory,denotes thejoinorleast upper boundoperation.; 3. Intopology,denotes thewedge sumof twopointed spaces.
$\land$ (wedge): 1. Denotes thelogical and,and is read as "and". If $E$ and $F$ arelogical 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. Inlattice theory,denotes themeetorgreatest lower boundoperation.; 3. Inmultilinear algebra,geometry,andmultivariable calculus,denotes thewedge productor theexterior product.
$⊻$: Exclusive or:if $E$ and $F$ are twoBoolean variablesorpredicates, $E\veebar F$ denotes the exclusive or. Notations $E XOR F$ and $E\oplus F$ are also commonly used; see⊕.
$\forall$ (turned A): 1. Denotesuniversal quantificationand is read as "for all". If $E$ is alogical predicate, $\forall x\;E$ means that $E$ is true for all possible values of the variable $x$ .; 2. Often used in plain text as an abbreviation of "for all" or "for every".
$\exists$: 1. Denotesexistential quantificationand is read "there exists... such that". If $E$ is alogical predicate, $\exists x\;E$ means that there exists at least one value of $x$ for which $E$ is true.; 2. Often used in plain text as an abbreviation of "there exists".
$\exists!$: Denotesuniqueness quantification,that is, $\exists!x\;P$ means "there exists exactly one $x$ such that $P$ (is true) ". In other words, $\exists!x\;P(x)$ is an abbreviation of $\exists x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x))$ .
$\Rightarrow$: 1. Denotesmaterial conditional,and is read as "implies". If $P$ and $Q$ arelogical 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 in plain text as an abbreviation of "implies".
$\Leftrightarrow$: 1. Denoteslogical equivalence,and is read "is equivalent to" or "if and only if".If $P$ and $Q$ arelogical 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 in plain text as an abbreviation of "if and only if".
$⊤$ (tee): 1. $\top$ denotes thelogical predicatealways true.; 2. Denotes also thetruth valuetrue.; 3. Sometimes denotes thetop elementof abounded lattice(previous meanings are specific examples).; 4. For the use as a superscript, see $□ ⊤$ .
$⊥$ (up tack): 1. $\bot$ denotes thelogical predicatealways false.; 2. Denotes also thetruth valuefalse.; 3. Sometimes denotes thebottom elementof abounded lattice(previous meanings are specific examples).; 4. InCryptographyoften denotes an error in place of a regular value.; 5. For the use as a superscript, see $□ ⊥$ .; 6. For the similar symbol, see $\perp$ .

Blackboard bold

Theblackboard bold typefaceis widely used for denoting the basicnumber 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 $\mathbb {R}$ incombinatorics,one should immediately know that this denotes thereal numbers,although combinatorics does not study the real numbers (but it uses them for many proofs).

$\mathbb {N}$: Denotes the set ofnatural numbers $\{1,2,\ldots \},$ or sometimes $\{0,1,2,\ldots \}.$ When the distinction is important and readers might assume either definition, $\mathbb {N} _{1}$ and $\mathbb {N} _{0}$ are used, respectively, to denote one of them unambiguously. Notation $\mathbf {N}$ is also commonly used.
$\mathbb {Z}$: Denotes the set ofintegers $\{\ldots,-2,-1,0,1,2,\ldots \}.$ It is often denoted also by $\mathbf {Z}.$
$\mathbb {Z} _{p}$: 1. Denotes the set of $p$ -adic integers,where $p$ is aprime number.; 2. Sometimes, $\mathbb {Z} _{n}$ denotes theintegers modulo $n$ ,where $n$ is anintegergreater than 0. The notation $\mathbb {Z} /n\mathbb {Z}$ is also used, and is less ambiguous.
$\mathbb {Q}$: Denotes the set ofrational 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 aprime number.
$\mathbb {R}$: Denotes the set ofreal numbers.It is often denoted also by $\mathbf {R}.$
$\mathbb {C}$: Denotes the set ofcomplex numbers.It is often denoted also by $\mathbf {C}.$
$\mathbb {H}$: Denotes the set ofquaternions.It is often denoted also by $\mathbf {H}.$
$\mathbb {F} _{q}$: Denotes thefinite fieldwith $q$ elements, where $q$ is aprime power(includingprime numbers). It is denoted also by $GF(q)$ .
$\mathbb {O}$: Used on rare occasions to denote the set ofoctonions.It is often denoted also by $\mathbf {O}.$

Calculus

$□'$: Lagrange's notationfor thederivative:If $f$ is afunctionof a single variable, $f'$ ,read as "fprime",is the derivative of $f$ with respect to this variable. Thesecond derivativeis the derivative of $f'$ ,and is denoted $f''$ .
${\dot {\Box }}$: Newton's notation,most commonly used for thederivativewith respect to time. If $x$ is a variable depending on time, then ${\dot {x}},$ read as "x dot", is its derivative with respect to time. In particular, if $x$ represents a moving point, then ${\dot {x}}$ is itsvelocity.
${\ddot {\Box }}$: Newton's notation,for thesecond derivative:If $x$ is a variable that represents a moving point, then ${\ddot {x}}$ is itsacceleration.
$.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠d □/d □⁠$: Leibniz's notationfor thederivative,which is used in several slightly different ways.; 1. If $y$ is a variable thatdependson $x$ ,then $\textstyle {\frac {\mathrm {d} y}{\mathrm {d} x}}$ ,read as "d y over d x" (commonly shortened to "d y d x" ), is the derivative of $y$ with respect to $x$ .; 2. If $f$ is afunctionof a single variable $x$ ,then $\textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}$ is the derivative of $f$ ,and $\textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}(a)$ is the value of the derivative at $a$ .; 3.Total derivative:If $f(x_{1},\ldots,x_{n})$ is afunctionof several variables thatdependon $x$ ,then $\textstyle {\frac {\mathrm {d} f}{\mathrm {d} x}}$ is the derivative of $f$ considered as a function of $x$ .That is, $\textstyle {\frac {\mathrm {d} f}{dx}}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}\,{\frac {\mathrm {d} x_{i}}{\mathrm {d} x}}$ .
$⁠ \partial □ / \partial □ ⁠$: Partial derivative:If $f(x_{1},\ldots,x_{n})$ is afunctionof several variables, $\textstyle {\frac {\partial f}{\partial x_{i}}}$ is the derivative with respect to the $i$ th variable considered as anindependent variable,the other variables being considered as constants.
$⁠ 𝛿 □ / 𝛿 □ ⁠$: Functional derivative:If $f(y_{1},\ldots,y_{n})$ is afunctionalof severalfunctions, $\textstyle {\frac {\delta f}{\delta y_{i}}}$ is the functional derivative with respect to the $n$ th function considered as anindependent variable,the other functions being considered constant.
${\overline {\Box }}$: 1.Complex conjugate:If $z$ is acomplex number,then ${\overline {z}}$ is its complex conjugate. For example, ${\overline {a+bi}}=a-bi$ .; 2.Topological closure:If $S$ is asubsetof atopological space $T$ ,then ${\overline {S}}$ is its topological closure, that is, the smallestclosed subsetof $T$ that contains $S$ .; 3.Algebraic closure:If $F$ is afield,then ${\overline {F}}$ is its algebraic closure, that is, the smallestalgebraically closed fieldthat contains $F$ .For example, ${\overline {\mathbb {Q} }}$ is the field of allalgebraic numbers.; 4.Mean value:If $x$ is avariablethat takes its values in some sequence of numbers $S$ ,then ${\overline {x}}$ may denote the mean of the elements of $S$ .; 5.Negation:Sometimes used to denote negation of the entire expression under the bar, particularly when dealing withBoolean algebra.For example, one ofDe Morgan's lawssays that ${\overline {A\land B}}={\overline {A}}\lor {\overline {B}}$ .
$\to$: 1. $A\to B$ denotes afunctionwithdomain $A$ andcodomain $B$ .For naming such a function, one writes $f:A\to B$ ,which is read as " $f$ from $A$ to $B$ ".; 2. More generally, $A\to B$ denotes ahomomorphismor amorphismfrom $A$ to $B$ .; 3. May denote alogical implication.For thematerial implicationthat is widely used in mathematics reasoning, it is nowadays generally replaced by⇒.Inmathematical logic,it remains used for denoting implication, but its exact meaning depends on the specific theory that is studied.; 4. Over avariable name,means that the variable represents avector,in a context where ordinary variables representscalars;for example, ${\overrightarrow {v}}$ .Boldface ( $\mathbf {v}$ ) or acircumflex( ${\hat {v}}$ ) are often used for the same purpose.; 5. InEuclidean geometryand more generally inaffine geometry, ${\overrightarrow {PQ}}$ denotes thevectordefined by the two points $P$ and $Q$ ,which can be identified with thetranslationthat maps $P$ to $Q$ .The same vector can be denoted also $Q-P$ ;seeAffine space.
$\mapsto$: "Maps to":Used for defining afunctionwithout having to name it. For example, $x\mapsto x^{2}$ is thesquare function.
$○$ ^[3]: 1.Function composition:If $f$ and $g$ are two functions, then $g\circ f$ is the function such that $(g\circ f)(x)=g(f(x))$ for every value of $x$ .; 2.Hadamard product of matrices:If $A$ and $B$ are two matrices of the same size, then $A\circ B$ is the matrix such that $(A\circ B)_{i,j}=(A)_{i,j}(B)_{i,j}$ .Possibly, $\circ$ is also used instead of $⊙$ for theHadamard product of power series.^{[citation needed]}
$\partial$: 1.Boundaryof atopological subspace:If $S$ is a subspace of a topological space, then itsboundary,denoted $\partial S$ ,is theset differencebetween theclosureand theinteriorof $S$ .; 2.Partial derivative:see $⁠ \partial□ / \partial□ ⁠$ .
$\int$: 1. Without a subscript, denotes anantiderivative.For example, $\textstyle \int x^{2}dx={\frac {x^{3}}{3}}+C$ .; 2. With a subscript and a superscript, or expressions placed below and above it, denotes adefinite integral.For example, $\textstyle \int _{a}^{b}x^{2}dx={\frac {b^{3}-a^{3}}{3}}$ .; 3. With a subscript that denotes a curve, denotes aline integral.For example, $\textstyle \int _{C}f=\int _{a}^{b}f(r(t))r'(t)\operatorname {d} t$ ,if $r$ is a parametrization of the curve $C$ ,from $a$ to $b$ .
$\oint$: Often used, typically in physics, instead of $\textstyle \int$ forline integralsover aclosed curve.
$\iint, ∯$: Similar to $\textstyle \int$ and $\textstyle \oint$ forsurface integrals.
${\boldsymbol {\nabla }}$ or ${\vec {\nabla }}$: Nabla,thegradient,vector derivative operator $\textstyle \left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)$ ,also calleddelorgrad,
$\nabla 2$ or $\nabla\cdot\nabla$: Laplace operatororLaplacian: $\textstyle {\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}$ .The forms $\nabla ^{2}$ and ${\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }}$ represent the dot product of thegradient( ${\boldsymbol {\nabla }}$ or ${\vec {\nabla }}$ ) with itself. Also notated $Δ$ (next item).
$Δ$: 1. Another notation for theLaplacian(see above).; 2. Operator offinite difference.
${\boldsymbol {\partial }}$ or $\partial _{\mu }$: Quad,the4-vector gradient operatororfour-gradient, $\textstyle \left({\frac {\partial }{\partial t}},{\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)$ .
$\Box$ or ${\Box }^{2}$: Denotes thed'Alembertianor squaredfour-gradient,which is a generalization of theLaplacianto four-dimensional spacetime. In flat spacetime with Euclidean coordinates, this may mean either $~\textstyle -{\frac {\partial ^{2}}{\partial t^{2}}}+{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}~\;$ or $\;~\textstyle +{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}~\;$ ;the sign convention must be specified. In curved spacetime (or flat spacetime with non-Euclidean coordinates), the definition is more complicated. Also calledboxorquabla.

Linear and multilinear algebra

$\sum$ (capital-sigma notation): 1. Denotes thesumof a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in $\textstyle \sum _{i=1}^{n}i^{2}$ or $\textstyle \sum _{0<i<j<n}j-i$ .; 2. Denotes aseriesand, if the series isconvergent,thesum of the series.For example, $\textstyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}$ .
$\prod$ (capital-pi notation): 1. Denotes theproductof a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in $\textstyle \prod _{i=1}^{n}i^{2}$ or $\textstyle \prod _{0<i<j<n}j-i$ .; 2. Denotes aninfinite product.For example, theEuler product formula for the Riemann zeta functionis $\textstyle \zeta (z)=\prod _{n=1}^{\infty }{\frac {1}{1-p_{n}^{-z}}}$ .; 3. Also used for theCartesian productof any number of sets and thedirect productof any number ofmathematical structures.
$\oplus$: 1. Internaldirect sum:if $E$ and $F$ areabelian subgroupsof anabelian group $V$ ,notation $V=E\oplus F$ means that $V$ is the direct sum of $E$ and $F$ ;that is, every element of $V$ can be written in a unique way as the sum of an element of $E$ and an element of $F$ .This applies also when $E$ and $F$ arelinear subspacesorsubmodulesof thevector spaceormodule $V$ .; 2.Direct sum:if $E$ and $F$ are twoabelian groups,vector spaces,ormodules,then their direct sum, denoted $E\oplus F$ is an abelian group, vector space, or module (respectively) equipped with twomonomorphisms $f:E\to E\oplus F$ and $g:F\to E\oplus F$ such that $E\oplus F$ is the internal direct sum of $f(E)$ and $g(F)$ .This definition makes sense because this direct sum is unique up to a uniqueisomorphism.; 3.Exclusive or:if $E$ and $F$ are twoBoolean variablesorpredicates, $E\oplus F$ may denote the exclusive or. Notations $E XOR F$ and $E\veebar F$ are also commonly used; see⊻.
$\otimes$: 1. Denotes thetensor productofabelian groups,vector spaces,modules,or other mathematical structures, such as in $E\otimes F,$ or $E\otimes _{K}F.$; 2. Denotes thetensor productof elements: if $x\in E$ and $y\in F,$ then $x\otimes y\in E\otimes F.$
$□ ⊤$: 1.Transpose:if $A$ is a matrix, $A^{\top }$ denotes thetransposeof $A$ ,that is, the matrix obtained by exchanging rows and columns of $A$ .Notation $^{\top }\!\!A$ is also used. The symbol $\top$ is often replaced by the letter $T$ or $t$ .; 2. For inline uses of the symbol, see⊤.
$□ ⊥$: 1.Orthogonal complement:If $W$ is alinear subspaceof aninner product space $V$ ,then $W^{\bot }$ denotes itsorthogonal complement,that is, the linear space of the elements of $V$ whose inner products with the elements of $W$ are all zero.; 2.Orthogonal subspacein thedual space:If $W$ is alinear subspace(or asubmodule) of avector space(or of amodule) $V$ ,then $W^{\bot }$ may denote theorthogonal subspaceof $W$ ,that is, the set of alllinear formsthat map $W$ to zero.; 3. For inline uses of the symbol, see⊥.

Advanced group theory

$⋉$ $⋊$: 1. Innersemidirect product:if $N$ and $H$ are subgroups of agroup $G$ ,such that $N$ is anormal subgroupof $G$ ,then $G=N\rtimes H$ and $G=H\ltimes N$ mean that $G$ is the semidirect product of $N$ and $H$ ,that is, that every element of $G$ can be uniquely decomposed as the product of an element of $N$ and an element of $H$ .(Unlike for thedirect product of groups,the element of $H$ may change if the order of the factors is changed.); 2. Outersemidirect product:if $N$ and $H$ are twogroups,and $\varphi$ is agroup homomorphismfrom $N$ to theautomorphism groupof $H$ ,then $N\rtimes _{\varphi }H=H\ltimes _{\varphi }N$ denotes a group $G$ ,unique up to agroup isomorphism,which is a semidirect product of $N$ and $H$ ,with the commutation of elements of $N$ and $H$ defined by $\varphi$ .
$≀$: Ingroup theory, $G\wr H$ denotes thewreath productof thegroups $G$ and $H$ .It is also denoted as $G\operatorname {wr} H$ or $G\operatorname {Wr} H$ ;seeWreath product § Notation and conventionsfor several notation variants.

Infinite numbers

$\infty$ (infinity symbol): 1. The symbol is read asinfinity.As an upper bound of asummation,aninfinite product,anintegral,etc., means that the computation is unlimited. Similarly, $-\infty$ in a lower bound means that the computation is not limited toward negative values.; 2. $-\infty$ and $+\infty$ are the generalized numbers that are added to thereal lineto form theextended real line.; 3. $\infty$ is the generalized number that is added to the real line to form theprojectively extended real line.
${\mathfrak {c}}$ (fraktur𝔠): ${\mathfrak {c}}$ denotes thecardinality of the continuum,which is thecardinalityof the set ofreal numbers.
$\aleph$ (aleph): With anordinal $i$ as a subscript, denotes the $i$ thaleph number,that is the $i$ th infinitecardinal.For example, $\aleph _{0}$ is the smallest infinite cardinal, that is, the cardinal of the natural numbers.
$\beth$ (bet (letter)): With anordinal $i$ as a subscript, denotes the $i$ thbeth number.For example, $\beth _{0}$ is thecardinalof the natural numbers, and $\beth _{1}$ is thecardinal of the continuum.
$\omega$ (omega): 1. Denotes the firstlimit ordinal.It is also denoted $\omega _{0}$ and can be identified with theordered setof thenatural numbers.; 2. With anordinal $i$ as a subscript, denotes the $i$ thlimit ordinalthat has acardinalitygreater than that of all preceding ordinals.; 3. Incomputer science,denotes the (unknown) greatest lower bound for the exponent of thecomputational complexityofmatrix multiplication.; 4. Written as afunctionof another function, it is used for comparing theasymptotic growthof two functions. SeeBig O notation § Related asymptotic notations.; 5. Innumber theory,may denote theprime omega function.That is, $\omega (n)$ is the number of distinct prime factors of the integer $n$ .

Brackets

Many sorts ofbracketsare 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 $□$ is used as a placeholder for schematizing the syntax that underlies the meaning.

Parentheses

$(□)$: Used in anexpressionfor specifying that the sub-expression between the parentheses has to be considered as a single entity; typically used for specifying theorder of operations.
$□(□)$ $□(□, □)$ $□(□,..., □)$: 1.Functional notation:if the first $\Box$ is the name (symbol) of afunction,denotes the value of the function applied to the expression between the parentheses; for example, $f(x)$ , $\sin(x+y)$ .In the case of amultivariate function,the parentheses contain several expressions separated by commas, such as $f(x,y)$ .; 2. May also denote a product, such as in $a(b+c)$ .When the confusion is possible, the context must distinguish which symbols denote functions, and which ones denotevariables.
$(□, □)$: 1. Denotes anordered pairofmathematical objects,for example, $(\pi,0)$ .; 2. If $a$ and $b$ arereal numbers, $-\infty$ ,or $+\infty$ ,and $a < b$ ,then $(a,b)$ denotes theopen intervaldelimited by $a$ and $b$ .See $]□, □[$ for an alternative notation.; 3. If $a$ and $b$ areintegers, $(a,b)$ may denote thegreatest common divisorof $a$ and $b$ .Notation $\gcd(a,b)$ is often used instead.
$(□, □, □)$: If $x, y, z$ are vectors in $\mathbb {R} ^{3}$ ,then $(x,y,z)$ may denote thescalar triple product.^{[citation needed]}See also[□,□,□]in§ Square brackets.
$(□,..., □)$: Denotes atuple.If there are $n$ objects separated by commas, it is an $n$ -tuple.
$(□, □,...)$ $(□,..., □,...)$: Denotes aninfinite sequence.
${\begin{pmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{pmatrix}}$: Denotes amatrix.Often denoted withsquare brackets.
${\binom {\Box }{\Box }}$: Denotes abinomial coefficient:Given twononnegative integers, ${\binom {n}{k}}$ is read as " $n$ choose $k$ ",and is defined as the integer ${\frac {n(n-1)\cdots (n-k+1)}{1\cdot 2\cdots k}}={\frac {n!}{k!\,(n-k)!}}$ (if $k = 0$ ,its value is conventionally $1$ ). Using the left-hand-side expression, it denotes apolynomialin $n$ ,and is thus defined and used for anyrealorcomplexvalue of $n$ .
$\left({\frac {\Box }{\Box }}\right)$: Legendre symbol:If $p$ is an oddprime numberand $a$ is aninteger,the value of $\left({\frac {a}{p}}\right)$ is 1 if $a$ is aquadratic residuemodulo $p$ ;it is –1 if $a$ is aquadratic non-residuemodulo $p$ ;it is 0 if $p$ divides $a$ .The same notation is used for theJacobi symbolandKronecker symbol,which are generalizations where $p$ is respectively any odd positive integer, or any integer.

Square brackets

$[□]$: 1. Sometimes used as a synonym of $(□)$ for avoiding nested parentheses.; 2.Equivalence class:given anequivalence relation, $[x]$ often denotes the equivalence class of the element $x$ .; 3.Integral part:if $x$ is areal number, $[x]$ often denotes the integral part ortruncationof $x$ ,that is, the integer obtained by removing all digits after thedecimal mark.This notation has also been used for other variants offloor and ceiling functions.; 4.Iverson bracket:if $P$ is apredicate, $[P]$ may denote the Iverson bracket, that is thefunctionthat takes the value $1$ for the values of thefree variablesin $P$ for which $P$ is true, and takes the value $0$ otherwise. For example, $[x=y]$ is theKronecker delta function,which equals one if $x=y$ ,and zero otherwise.; 5. In combinatorics or computer science, sometimes $[n]$ with $n\in \mathbb {N}$ denotes the set $\{1,2,3,\ldots,n\}$ of positive integers up to $n$ ,with $[0]=\emptyset$ .
$□[□]$: Image of a subset:if $S$ is asubsetof thedomain of the function $f$ ,then $f[S]$ is sometimes used for denoting the image of $S$ .When no confusion is possible, notation $f (S)$ is commonly used.
$[□, □]$: 1.Closed interval:if $a$ and $b$ arereal numberssuch that $a\leq b$ ,then $[a,b]$ denotes the closed interval defined by them.; 2.Commutator (group theory):if $a$ and $b$ belong to agroup,then $[a,b]=a^{-1}b^{-1}ab$ .; 3.Commutator (ring theory):if $a$ and $b$ belong to aring,then $[a,b]=ab-ba$ .; 4. Denotes theLie bracket,the operation of aLie algebra.
$[□: □]$: 1.Degree of a field extension:if $F$ is anextensionof afield $E$ ,then $[F:E]$ denotes the degree of thefield extension $F/E$ .For example, $[\mathbb {C}:\mathbb {R} ]=2$ .; 2.Index of a subgroup:if $H$ is asubgroupof agroup $E$ ,then $[G:H]$ denotes the index of $H$ in $G$ .The notation $| G:H |$ is also used
$[□, □, □]$: If $x, y, z$ are vectors in $\mathbb {R} ^{3}$ ,then $[x,y,z]$ may denote thescalar triple product.^[4]See also(□,□,□)in§ Parentheses.
${\begin{bmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{bmatrix}}$: Denotes amatrix.Often denoted withparentheses.

Braces

${ }$: Set-builder notationfor theempty set,also denoted $\emptyset$ or∅.
${□}$: 1. Sometimes used as a synonym of $(□)$ and $[□]$ for avoiding nested parentheses.; 2.Set-builder notationfor asingleton set: $\{x\}$ denotes thesetthat has $x$ as a single element.
${□,..., □}$: Set-builder notation:denotes thesetwhose elements are listed between the braces, separated by commas.
${□: □}$ ${□ | □}$: Set-builder notation:if $P(x)$ is apredicatedepending on avariable $x$ ,then both $\{x:P(x)\}$ and $\{x\mid P(x)\}$ denote thesetformed by the values of $x$ for which $P(x)$ is true.
Single brace: 1. Used for emphasizing that severalequationshave to be considered assimultaneous equations;for example, $\textstyle {\begin{cases}2x+y=1\\3x-y=1\end{cases}}$ .; 2.Piecewisedefinition; for example, $\textstyle |x|={\begin{cases}x&{\text{if }}x\geq 0\\-x&{\text{if }}x<0\end{cases}}$ .; 3. Used for grouped annotation of elements in a formula; for example, $\textstyle \underbrace {(a,b,\ldots,z)} _{26}$ , $\textstyle \overbrace {1+2+\cdots +100} ^{=5050}$ , $\textstyle \left.{\begin{bmatrix}A\\B\end{bmatrix}}\right\}m+n{\text{ rows}}$

Other brackets

$|□|$

1.Absolute value:if

x

is arealorcomplexnumber,

|x|

denotes its absolute value.

2. Number of elements: If

S

is aset,

|S|

may denote itscardinality,that is, its number of elements.

\#S

is also often used, see

#

.

3. Length of aline segment:If

P

and

Q

are two points in aEuclidean space,then

|PQ|

often denotes the length of the line segment that they define, which is thedistancefrom

P

to

Q

,and is often denoted

d(P,Q)

.

4. For a similar-looking operator, see

|

.

$| □:□ |$

Index of a subgroup:if

H

is asubgroupof agroup

G

,then

|G:H|

denotes the index of

H

in

G

.The notation

[G:H]

is also used

$\textstyle {\begin{vmatrix}\Box &\cdots &\Box \\\vdots &\ddots &\vdots \\\Box &\cdots &\Box \end{vmatrix}}$

{\begin{vmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{vmatrix}}

denotes thedeterminantof thesquare matrix

{\begin{bmatrix}x_{1,1}&\cdots &x_{1,n}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,n}\end{bmatrix}}

.

$||□||$

1. Denotes thenormof an element of anormed vector space.

2. For the similar-looking operator namedparallel,see

∥

.

$⌊□⌋$

Floor function:if

x

is a real number,

\lfloor x\rfloor

is the greatestintegerthat is not greater than

x

.

$⌈□⌉$

Ceiling function:if

x

is a real number,

\lceil x\rceil

is the lowestintegerthat is not lesser than

x

.

$⌊□⌉$

Nearest integer function:if

x

is a real number,

\lfloor x\rceil

is theintegerthat is the closest to

x

.

$]□, □[$

Open interval:If a and b are real numbers,

-\infty

,or

+\infty

,and

a<b

,then

]a,b[

denotes the open interval delimited by a and b. See

(□, □)

for an alternative notation.

$(□, □]$ $]□, □]$

Both notations are used for aleft-open interval.

$[□, □)$ $[□, □[$

Both notations are used for aright-open interval.

$⟨□⟩$

1.Generated object:if

S

is a set of elements in an algebraic structure,

\langle S\rangle

denotes often the object generated by

S

.If

S=\{s_{1},\ldots,s_{n}\}

,one writes

\langle s_{1},\ldots,s_{n}\rangle

(that is, braces are omitted). In particular, this may denote

thelinear spanin avector space(also often denoted $Span(S)$ ),
the generatedsubgroupin agroup,
the generatedidealin aring,
the generatedsubmodulein amodule.

2. Often used, mainly in physics, for denoting anexpected value.Inprobability theory,

E(X)

is generally used instead of

\langle S\rangle

.

$⟨□, □⟩$ $⟨□ | □⟩$

Both

\langle x,y\rangle

and

\langle x\mid y\rangle

are commonly used for denoting theinner productin aninner product space.

$\langle \Box |{\text{ and }}|\Box \rangle$

Bra–ket notationorDirac notation:if

x

and

y

are elements of aninner product space,

|x\rangle

is the vector defined by

x

,and

\langle y|

is thecovectordefined by

y

;their inner product is

\langle y\mid x\rangle

.

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 inclassical logicfor 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 ablack boardfor indicating relationships between formulas.

$■, □$: Used for marking the end of a proof and separating it from the current text. Theinitialism Q.E.D. or QED(Latin: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.
$☡$: 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.
$∴$: 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."
$∵$: Abbreviation of "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: " $11$ isprime∵ it has no positive integer factors other than itself and one. "
$∋$: 1. Abbreviation of "such that". For example, $x\ni x>3$ is normally printed " $x$ such that $x>3$ ".; 2. Sometimes used for reversing the operands of $\in$ ;that is, $S\ni x$ has the same meaning as $x\in S$ .See∈in§ Set theory.
$\propto$: Abbreviation of "is proportional to".

Miscellaneous

$!$: 1.Factorial:if $n$ is apositive integer, $n!$ is the product of the first $n$ positive integers, and is read as "n factorial".; 2.Double factorial:if $n$ is apositive integer, $n!!$ is the product of all positive integers up to $n$ with the same parity as $n$ ,and is read as "the double factorial of n".; 3.Subfactorial:if $n$ is a positive integer, $! n$ is the number ofderangementsof a set of $n$ elements, and is read as "the subfactorial of n".
$*$: Many different uses in mathematics; seeAsterisk § Mathematics.
$|$: 1.Divisibility:if $m$ and $n$ are two integers, $m\mid n$ means that $m$ divides $n$ evenly.; 2. Inset-builder notation,it is used as a separator meaning "such that"; see ${□ | □}$ .; 3.Restriction of a function:if $f$ is afunction,and $S$ is asubsetof itsdomain,then $f|_{S}$ is the function with $S$ as a domain that equals $f$ on $S$ .; 4.Conditional probability: $P(X\mid E)$ denotes the probability of $X$ given that the event $E$ occurs. Also denoted $P(X/E)$ ;see "/".; 5. For several uses asbrackets(in pairs or with $⟨$ and $⟩$ ) see§ Other brackets.
$∤$: Non-divisibility: $n\nmid m$ means that $n$ is not a divisor of $m$ .
$∥$: 1. Denotesparallelisminelementary geometry:if $PQ$ and $RS$ are twolines, $PQ\parallel RS$ means that they are parallel.; 2.Parallel,anarithmetical operationused inelectrical engineeringfor modelingparallel resistors: $x\parallel y={\frac {1}{{\frac {1}{x}}+{\frac {1}{y}}}}$ .; 3. Used in pairs as brackets, denotes anorm;see $||□||$ .; 4.Concatenation:Typically used in computer science, $x\mathbin {\vert \vert } y$ is said to represent the value resulting from appending the digits of $y$ to the end of $x$ .; 5. ${\displaystyle D_{\text{KL}}(P\parallel Q)}$ ,denotes astatistical distanceor measure of how oneprobability distributionP is different from a second, reference probability distribution Q.
$∦$: Sometimes used for denoting that twolinesare not parallel; for example, $PQ\not \parallel RS$ .
$\perp$: 1. Denotesperpendicularityandorthogonality.For example, if $A, B, C$ are three points in aEuclidean space,then $AB\perp AC$ means that theline segments $AB$ and $AC$ areperpendicular,and form aright angle.; 2. For the similar symbol, see $\bot$ .
$⊙$: Hadamard product of power series:if $\textstyle S=\sum _{i=0}^{\infty }s_{i}x^{i}$ and $\textstyle T=\sum _{i=0}^{\infty }t_{i}x^{i}$ ,then $\textstyle S\odot T=\sum _{i=0}^{\infty }s_{i}t_{i}x^{i}$ .Possibly, $\odot$ is also used instead of $○$ for theHadamard product of matrices.^{[citation needed]}

^ISO 80000-2,Section 9 "Operations", 2-9.6
^"Statistics and Data Analysis: From Elementary to Intermediate".
^TheLaTeXequivalent to bothUnicodesymbols ∘ 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 aninterpoint,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.
^Rutherford, D. E. (1965).Vector Methods.University Mathematical Texts. Oliver and Boyd Ltd., Edinburgh.

External links

Some Unicode charts of mathematical operators and symbols:

Some Unicode cross-references:

Short list of commonly used LaTeX symbolsandComprehensive LaTeX Symbol List
MathML Characters- sorts out Unicode, HTML and MathML/TeX names on one page
Unicode values and MathML names
Unicode values and Postscript namesfrom the source code forGhostscript

[ISO-1] ISO 80000-2,Section 9 "Operations", 2-9.6

[2] "Statistics and Data Analysis: From Elementary to Intermediate".

[3] TheLaTeXequivalent to bothUnicodesymbols ∘ 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 aninterpoint,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.

[4] Rutherford, D. E. (1965).Vector Methods.University Mathematical Texts. Oliver and Boyd Ltd., Edinburgh.

[1]

[2]

[3]

[4]

Glossary of mathematical symbols

Contents

Layout of this article

Arithmetic operators

Equality, equivalence and similarity

Comparison

Set theory

Basic logic

Blackboard bold

Calculus

Linear and multilinear algebra

Advanced group theory

Infinite numbers

Brackets

Parentheses

Square brackets

Braces

Other brackets

Symbols that do not belong to formulas

Miscellaneous

See also

Related articles

Related lists

Unicodesymbols

References

External links