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

• A – adele ring or algebraic numbers.
• AC –
  Axiom of Choice,^[1] 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.^[2]
• 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.^[3]
• Bias – bias of an estimator .
• BWOC –
  by way of contradiction
  .

C

• C – complex numbers.
• Card – cardinality of a set.^[4] (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 ${\displaystyle \nabla }$.)
• 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.^[1]
  (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.^[2]
  (Also written as ${\displaystyle \Im }$.)
• 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.^[5]
• lg – common logarithm (log₁₀) or binary logarithm (log₂).
• LHS –
  left-hand side
  of an equation.
• Li –
  offset logarithmic integral
  function.
• li –
  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.^[6]
• LST –
  language of set theory
  .
• lub –
  least upper bound.^[1]
  (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.^[4]

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 ${\displaystyle \mathbb {P} }$.)
• 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.^[2]
  (Also written ${\displaystyle \Re }$.)
• 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.^[1]
• 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.^[1]
  (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.^[1]
• 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.^[4]
• ZFC –
  Axiom of Choice) of set theory.^[4]

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

References