# List of mathematical abbreviations

This following list features abbreviated names of mathematical functions, function-like operators and other mathematical terminology.
    This list is limited to abbreviations of two or more letters (excluding number sets). The capitalization of some of these abbreviations is not standardized – different authors might use different capitalizations.
    This list is incomplete; you can help by adding missing items. (February 2011)
* A
* B
* C
* D
* E
* F
* G
* H
* I
* K
* L
* M
* N
* O
* P
* Q
* R
* S
* T
* U
* V
* W
* X
* Z
* See also
* References

## A

* A – adele ring or algebraic numbers.
* a.a.s. – asymptotically almost surely.
* AC – Axiom of Choice, or set of absolutely continuous functions.
* a.c. – absolutely continuous.
* acrd – inverse chord function.
* ad – adjoint representation (or adjoint action) of a Lie group.
* adj – adjugate of a matrix.
* a.e. – almost everywhere.
* AFSOC - Assume for the sake of contradiction
* Ai – Airy function.
* AL – Action limit.
* Alt – alternating group (Alt(n) is also written as A$_n$.)
* A.M. – arithmetic mean.
* AP – arithmetic progression.
* arccos – inverse cosine function.
* arccosec – inverse cosecant function. (Also written as arccsc.)
* arccot – inverse cotangent function.
* arccsc – inverse cosecant function. (Also written as arccosec.)
* arcexc – inverse excosecant function. (Also written as arcexcsc, arcexcosec.)
* arcexcosec – inverse excosecant function. (Also written as arcexcsc, arcexc.)
* arcexcsc – inverse excosecant function. (Also written as arcexcosec, arcexc.)
* arcexs – inverse exsecant function. (Also written as arcexsec.)
* arcexsec – inverse exsecant function. (Also written as arcexs.)
* arcosech – inverse hyperbolic cosecant function. (Also written as arcsch.)
* arcosh – inverse hyperbolic cosine function.
* arcoth – inverse hyperbolic cotangent function.
* arcsch – inverse hyperbolic cosecant function. (Also written as arcosech.)
* arcsec – inverse secant function.
* arcsin – inverse sine function.
* arctan – inverse tangent function.
* arctan2 – inverse tangent function with two arguments. (Also written as atan2.)
* arg – argument of.
* arg max – argument of the maximum.
* arg min – argument of the minimum.
* arsech – inverse hyperbolic secant function.
* arsinh – inverse hyperbolic sine function.
* artanh – inverse hyperbolic tangent function.
* a.s. – almost surely.
* atan2 – inverse tangent function with two arguments. (Also written as arctan2.)
* A.P. – arithmetic progression.
* Aut – automorphism group.

## B

* bd – boundary. (Also written as fr or ∂.)
* Bi – Airy function of the second kind.
* BIDMAS – Brackets, Indices, Divide, Multiply, Add, Subtract.
* Bias – bias of an estimator .
* BWOC – by way of contradiction.

## C

* C – complex numbers.
* Card – cardinality of a set. (Card(X) is also written #X, ♯X or |X|.)
* cas – cos + sin function.
* cdf – cumulative distribution function.
* c.f. – cumulative frequency.
* c.c. – complex conjugate.
* char – characteristic of a ring.
* Chi – hyperbolic cosine integral function.
* Ci – cosine integral function.
* cis – cos + i sin function. (Also written as expi.)
* Cl – conjugacy class.
* cl – topological closure.
* CLT – central limit theorem.
* cod, codom – codomain.
* cok, coker – cokernel.
* colsp – column space of a matrix.
* conv – convex hull of a set.
* Cor – corollary.
* corr – correlation.
* cos – cosine function.
* cosec – cosecant function. (Also written as csc.)
* cosech – hyperbolic cosecant function. (Also written as csch.)
* cosh – hyperbolic cosine function.
* cosiv – coversine function. (Also written as cover, covers, cvs.)
* cot – cotangent function. (Also written as ctg.)
* coth – hyperbolic cotangent function.
* cov – covariance of a pair of random variables.
* cover – coversine function. (Also written as covers, cvs, cosiv.)
* covercos – covercosine function. (Also written as cvc.)
* covers – coversine function. (Also written as cover, cvs, cosiv.)
* crd – chord function.
* CRT – Chinese remainder theorem.
* csc – cosecant function. (Also written as cosec.)
* csch – hyperbolic cosecant function. (Also written as cosech.)
* ctg – cotangent function. (Also written as cot.)
* curl – curl of a vector field. (Also written as rot.)
* cvc – covercosine function. (Also written as covercos.)
* cvs – coversine function. (Also written as cover, covers, cosiv.)

## D

* def – define or definition.
* deg – degree of a polynomial, or other recursively-defined objects such as well-formed formulas. (Also written as ∂.)
* del – del, a differential operator. (Also written as $\nabla$.)
* det – determinant of a matrix or linear transformation.
* DFT – discrete Fourier transform.
* dim – dimension of a vector space.
* div – divergence of a vector field.
* DNE – a solution for an expression does not exist, or is undefined. Generally used with limits and integrals.
* dom, domain – domain of a function. (Or, more generally, a relation.)

## E

* End – categories of endomorphisms.
* Ei – exponential integral function.
* epi – epigraph of a function.
* Eqn – equation.
* erf – error function.
* erfc – complementary error function.
* erfcx – scaled complementary error function.
* erfi – imaginary error function.
* etr – exponent of the trace.
* exc – excosecant function. (Also written as excsc, excosec.)
* excosec – excosecant function. (Also written as excsc, exc.)
* excsc – excosecant function. (Also written as excosec, exc.)
* exs – exsecant function. (Also written as exsec.)
* exsec – exsecant function. (Also written as exs.)
* exp – exponential function. (exp x is also written as e$^x$.)
* expi – cos + i sin function. (Also written as cis.)
* expm1 – exponential minus 1 function. (Also written as exp1m.)
* exp1m – exponential minus 1 function. (Also written as expm1.)
* Ext – Ext functor.
* ext – exterior.
* extr – a set of extreme points of a set.

## F

* FFT – fast Fourier transform.
* FIP – finite intersection property.
* FOC – first order condition.
* FOL – first-order logic.
* fr – boundary. (Also written as bd or ∂.)
* Frob – Frobenius endomorphism.
* FT – Fourier transform.
* FTA – fundamental theorem of arithmetic or fundamental theorem of algebra.

## G

* Gal – Galois group. (Also written as Γ.)
* gcd – greatest common divisor of two numbers. (Also written as hcf.)
* gd – Gudermannian function.
* GF – Galois field.
* GF – generating function.
* GL – general linear group.
* G.M. – geometric mean.
* glb – greatest lower bound. (Also written as inf.)
* G.P. – geometric progression.
* grad – gradient of a function.
* graph – graph of a function.

## H

* H – quaternion numbers.
* hacover – hacoversine function. (Also written as hacovers, hcv.)
* hacovercos – hacovercosine function. (Also written as hcc.)
* hacovers – hacoversine function. (Also written as hacover, hcv.)
* hav – haversine function. (Also written as sem.)
* havercos – havercosine function. (Also written as hvc.)
* h.c. – Hermitian conjugate, often used as part of + h.c. (Also written as H.c.)
* hcc – hacovercosine function. (Also written as hacovercos.)
* hcv – hacoversine function. (Also written as hacover, hacovers.)
* hcf – highest common factor of two numbers. (Also written as gcd.)
* H.M. – harmonic mean.
* HOL – higher-order logic.
* Hom – Hom functor.
* hom – hom-class.
* hot – higher order term.
* HOTPO – half or triple plus one.
* hvc – havercosine function. (Also written as havercos.)
* hyp – hypograph of a function.

## I

* iff – if and only if.
* IH – induction hypothesis.
* iid – independent and identically distributed random variables.
* Im – imaginary part of a complex number. (Also written as $\Im$.)
* im – image.
* inf – infimum of a set. (Also written as glb.)
* int – interior.
* I.o. – Infinitely often.

## K

* ker – kernel.

## L

* lb – binary logarithm (log$_2$). (Also written as ld.)
* lcm – lowest common multiple (a.k.a. least common multiple) of two numbers.
* LCHS – locally compact Hausdorff second countable.
* ld – binary logarithm (log$_2$). (Also written as lb.)
* lsc – lower semi-continuity.
* lerp – linear interpolation.
* lg – common logarithm (log$_{10}$) or binary logarithm (log$_2$).
* LHS – left-hand side of an equation.
* Li – offset logarithmic integral function.
* li – logarithmic integral function or linearly independent.
* lim – limit of a sequence, or of a function.
* lim inf – limit inferior.
* lim sup – limit superior.
* LLN – law of large numbers.
* ln – natural logarithm, log$_e$.
* lnp1 – natural logarithm plus 1 function.
* ln1p – natural logarithm plus 1 function.
* log – logarithm. (If without a subscript, this may mean either log$_{10}$ or log$_e$.)
* logh – natural logarithm, log$_e$.
* LST – language of set theory.
* lub – least upper bound. (Also written sup.)

## M

* max – maximum of a set.
* MGF – moment-generating function.
* M.I. – mathematical induction.
* min – minimum of a set.
* mod – modulo.
* Mp – metaplectic group.
* mtanh – modified hyperbolic tangent function. (Also written as mth.)
* mth – modified hyperbolic tangent function. (Also written as mtanh.)
* mx – matrix.

## N

* N – natural numbers.
* NAND – not-and in logic.
* No. – number.
* NOR – not-or in logic.
* NTS – need to show.
* Null, null – (See Kernel.)
* Nullity, nullity – nullity.

## O

* O – octonion numbers.
* OBGF – ordinary bivariate generating function.
* ob – object class.
* ord – ordinal number of a well-ordered set.
* O/W - otherwise.

## P

* pdf – probability density function.
* pf – proof.
* PGL – projective general linear group.
* Pin – pin group.
* pmf – probability mass function.
* Pn – previous number.
* Pr – probability of an event. (See Probability theory. Also written as P or $\mathbb {P}$.)
* probit – probit function.
* PRNG – pseudorandom number generator.
* PSL – projective special linear group.
* PNT – prime number theorem.
* PRP – probable prime.
* PSO – projective orthogonal group.
* PSU – projective special unitary group.
* PU – projective unitary group.

## Q

* Q – rational numbers.
* QED – "Quod erat demonstrandum", a Latin phrase used at the end of a definitive proof.
* QEF – "Quod erat faciendum", a Latin phrase sometimes used at the end of a geometrical construction.

## R

* R – real numbers.
* ran – range of a function.
* rank – rank of a matrix. (Also written as rk.)
* Re – real part of a complex number. (Also written $\Re$.)
* resp – respectively.
* RHS – right-hand side of an equation.
* rk – rank. (Also written as rank.)
* RMS, rms – root mean square.
* rng – non-unital ring.
* rot – rotor of a vector field. (Also written as curl.)
* rowsp – row space of a matrix.
* RTP – required to prove.
* RV – random variable. (Also written as R.V.)

## S

* S – sedenion numbers.
* SD – standard deviation.
* SE – standard error.
* sec – secant function.
* sech – hyperbolic secant function.
* seg – initial segment of.
* sem – haversine function. (Also written as hav.)
* SFIP – strong finite intersection property.
* sgn – sign function.
* Shi – hyperbolic sine integral function.
* Si – sine integral function.
* sigmoid – sigmoid function.
* sin – sine function.
* sinc – sinc function.
* sinh – hyperbolic sine function.
* siv – versine function. (Also written as ver, vers.)
* SL – special linear group.
* SO – special orthogonal group.
* SOC – second order condition.
* Soln – solution.
* Sp – symplectic group.
* Sp – trace of a matrix, from the German "spur" used for the trace.
* sp, span – linear span of a set of vectors. (Also written with angle brackets.)
* Spec – spectrum of a ring.
* Spin – spin group.
* sqrt – square root.
* s.t. – such that or so that or subject to.
* st – standard part function.
* STP – [it is] sufficient to prove.
* SU – special unitary group.
* sup – supremum of a set. (Also written as lub, which stands for least upper bound.)
* supp – support of a function.
* swish – swish function, an activation function in data analysis.
* Sym – symmetric group (Sym(n) is also written as S$_n$) or symmetric algebra.

## T

* T – trigintaduonion numbers.
* tan – tangent function. (Also written as tgn, tg.)
* tanh – hyperbolic tangent function.
* TFAE – the following are equivalent.
* tg – tangent function. (Also written as tan, tgn.)
* tgn – tangent function. (Also written as tan, tg.)
* Thm – theorem.
* Tor – Tor functor.
* Tr – field trace.
* tr – trace of a matrix or linear transformation. (Also written as Sp.)

## U

* undef – a function or expression is undefined.
* usc – upper semi-continuity.

## V

* V – volume.
* var – variance of a random variable.
* vcs – vercosine function. (Also written as vercos.)
* ver – versine function. (Also written as vers, siv.)
* vercos – vercosine function. (Also written as vcs.)
* vers – versine function. (Also written as ver, siv.)

## W

* W^5 – which was what we wanted. Synonym of Q.E.D.
* walog – without any loss of generality.
* wff – well-formed formula.
* whp – with high probability.
* wlog – without loss of generality.
* WMA – we may assume.
* WO – well-ordered set.
* WOP – well-ordered principle.
* w.p. – with probability.
* wp1 – with probability 1.
* wrt – with respect to or with regard to.
* WTP – want to prove.
* WTS – want to show.

## X

* XOR – exclusive or in logic.

## Z

* Z – integer numbers.
* ZF – Zermelo–Fraenkel axioms of set theory.
* ZFC – Zermelo–Fraenkel axioms (with the Axiom of Choice) of set theory.