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.

A

• A – adele ring or algebraic numbers.
• 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 (log₂). (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 (log₂). (Also written as lb.)
• lsc – lower semi-continuity.
• lerp – linear interpolation.
• lg – common logarithm (log₁₀) or binary logarithm (log₂).
• 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 log₁₀ 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.

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

• 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.

See also

• List of letters used in mathematics, science, and engineering
• ISO 31-11
• Language of mathematics
• List of mathematical jargon
• Mathematical notation
• Notation in probability and statistics
• Physical constants
• List of logic symbols
• Glossary of mathematical symbols
• Mathematical operators and symbols in Unicode
• List of mathematical functions