Symbol Catalog
Browse the notation used across the practice problems. Each card opens a focused explanation page.
Limit
\limA limit describes the value an expression approaches as its input gets close to a target.
Derivative
\frac{\mathrm{d}}{\mathrm{d}x}A derivative measures instantaneous rate of change with respect to a variable.
Integral
\intAn integral accumulates signed area or reverses differentiation.
Prime-counting function
\pi\!(n)The prime-counting function gives the number of primes less than or equal to n.
For all
\forallThe universal quantifier means a statement applies to every object in a set.
Summation
\sumA summation adds a sequence of terms indexed over a range.
Exponential
e^xThe natural exponential function has growth rate equal to its own value.
QED
\mathrm{QED}QED marks the end of a proof, meaning the claim has been shown.
Modulo
\bmodModulo arithmetic compares remainders after division.
Square root
\sqrt{n}A square root is a value that squares to the quantity under the radical.
Prime notation
f'Prime notation is a compact way to denote derivatives of a function.
Infinity
\inftyInfinity represents unbounded growth or an endless process, not a regular number.
Greatest common divisor
\gcdThe greatest common divisor is the largest integer that divides two integers.
Therefore
\thereforeTherefore indicates that a conclusion follows from previous statements.
Natural logarithm
\lnThe natural logarithm is the inverse of the exponential function e^x.
Prime number
pA prime number has exactly two positive divisors: 1 and itself.
Partial derivative
\partialA partial derivative changes one variable while holding the others fixed.
Real numbers
\mathbb{R}The real numbers include rational and irrational quantities on the number line.
Exists
\existsThe existential quantifier means at least one object satisfies a statement.
Not equal
\neNot equal states that two expressions do not have the same value.
Approximately equal
\approxApproximately equal compares quantities that are close but not necessarily identical.
Less than or equal
\leLess than or equal allows equality or a smaller value.
Greater than or equal
\geGreater than or equal allows equality or a larger value.
Element of
\inElement of says that an object belongs to a set.
Not an element of
\notinNot an element of says that an object does not belong to a set.
Subset
\subsetA subset is a set whose elements are all contained in another set.
Union
\cupA union combines all elements that appear in either set.
Intersection
\capAn intersection keeps only the elements shared by both sets.
Empty set
\varnothingThe empty set contains no elements.
Gradient
\nablaThe gradient collects partial derivatives into a vector pointing in the direction of steepest increase.
Delta
\DeltaDelta often represents a finite change in a quantity.
Epsilon
\varepsilonEpsilon usually represents a small positive quantity, especially in limit proofs.
Theta
\thetaTheta commonly represents an angle or a parameter.
Lambda
\lambdaLambda is often used for eigenvalues, rates, or parameters.
Factorial
n!A factorial multiplies all positive integers up to n.
Absolute value
|x|Absolute value measures distance from zero on the number line.
Plus or minus
\pmPlus or minus represents two possible signs in one expression.
Implies
\RightarrowImplies indicates that one statement leads logically to another.
If and only if
\LeftrightarrowIf and only if means two statements imply each other.
Complex numbers
\mathbb{C}The complex numbers extend the real numbers with the imaginary unit i.
Integers
\mathbb{Z}The integers are whole numbers, including negative values and zero.
Rational numbers
\mathbb{Q}Rational numbers can be written as a ratio of two integers with nonzero denominator.
Natural numbers
\mathbb{N}The natural numbers are the counting numbers, usually starting at 1 or sometimes 0 depending on convention.
Subset or equal
\subseteqSubset or equal means every element of one set is in another set, and the two sets may be equal.
Superset
\supsetA superset contains every element of another set.
Superset or equal
\supseteqSuperset or equal means one set contains another set, with equality allowed.
Set difference
\setminusSet difference keeps elements from the first set that are not in the second set.
Complement
A^cThe complement contains all elements in the universe that are not in the set.
Power set
\mathcal{P}(A)The power set is the set of all subsets of a set.
Cartesian product
A\times BThe Cartesian product is the set of ordered pairs formed from two sets.
Set-builder notation
\{x\mid P(x)\}Set-builder notation describes a set by the property its elements satisfy.
Cardinality
|A|Cardinality counts how many elements are in a set.
Sine
\sinSine is a trigonometric function that relates an angle to a ratio in a right triangle or unit circle.
Cosine
\cosCosine is a trigonometric function that tracks horizontal coordinate on the unit circle.
Tangent
\tanTangent is the ratio of sine to cosine and also describes slope angle.
Product notation
\prodProduct notation multiplies a sequence of indexed factors.
Binomial coefficient
\binom{n}{k}A binomial coefficient counts the number of ways to choose k objects from n objects.
Floor
\lfloor x\rfloorThe floor function returns the greatest integer less than or equal to a value.
Ceiling
\lceil x\rceilThe ceiling function returns the least integer greater than or equal to a value.
Angle
\angleAngle notation marks the measure or object formed by two rays sharing a vertex.
Degree
^\circDegrees measure angles, with a full turn equal to 360 degrees.
Vector
\vec{v}A vector has both magnitude and direction.
Dot product
\cdotThe dot product multiplies two vectors to produce a scalar measuring alignment.
Cross product
\timesThe cross product creates a vector perpendicular to two three-dimensional vectors.
Matrix
\begin{bmatrix}a&b\\c&d\end{bmatrix}A matrix is a rectangular array of numbers or expressions.
Determinant
\detThe determinant is a scalar that captures scaling and invertibility information about a square matrix.
Transpose
A^TThe transpose flips a matrix across its main diagonal.
Probability
\mathbb{P}Probability measures how likely an event is to occur.
Expected value
\mathbb{E}Expected value is the long-run average value of a random variable.
Variance
\operatorname{Var}Variance measures how spread out a random variable is around its expected value.
Sigma
\sigmaLowercase sigma often denotes standard deviation or a parameter.
Proportional to
\proptoProportional to means one quantity is a constant multiple of another.
Equivalent
\equivEquivalent can express identity, logical equivalence, or congruence depending on context.
Parallel
\parallelParallel lines or vectors point in the same or exactly opposite directions.
Perpendicular
\perpPerpendicular objects meet at a right angle.
Alpha
\alphaAlpha is a Greek letter often used for angles, parameters, and coefficients.
Beta
\betaBeta is a Greek letter commonly used for angles, parameters, and special functions.
Gamma
\gammaGamma is used for constants, angles, and the gamma function.
Mu
\muMu often represents a mean, measure, or parameter.
Rho
\rhoRho often denotes density, correlation, or a radial coordinate.
Tau
\tauTau is used for time constants, angles, and sometimes a full-turn constant.
Omega
\omegaOmega often represents angular velocity, frequency, or an outcome.
Capital omega
\OmegaCapital omega can represent a sample space, domain, or asymptotic lower bound.
Psi
\psiPsi is often used for wavefunctions or special functions.
Phi
\phiPhi can represent an angle, a function, or the golden ratio depending on context.
Chi
\chiChi appears in characteristic functions, statistics, and Greek-indexed notation.
Eta
\etaEta is a Greek letter used for parameters, efficiency, or small variables.
Xi
\xiXi is often used as a variable in analysis, probability, and special functions.
Not
\negLogical not negates a proposition.
And
\landLogical and is true when both propositions are true.
Or
\lorLogical or is true when at least one proposition is true.
Turnstile
\vdashTurnstile denotes syntactic entailment or provability.
Models
\modelsModels denotes semantic entailment or satisfaction.
Similar
\simSimilar can indicate asymptotic equivalence, geometric similarity, or a distribution relation.
Congruent
\congCongruent means equivalent in shape or equivalent under a relation.
Isomorphic
\simeqIsomorphic objects have the same structure even if their labels differ.
Asymptotic comparison
\asympAsymptotic comparison says two quantities have comparable growth up to constants.
Maps to
\mapstoMaps to shows where a function sends a specific input.
Function arrow
\toThe arrow can describe mappings, limits, or transitions.
Composition
\circComposition applies one function after another.
Plus
+Plus denotes addition or a positive sign.
Minus
-Minus denotes subtraction or a negative sign.
Division
\divDivision splits a quantity into equal parts or forms a ratio.
Fraction
\frac{a}{b}A fraction represents division of a numerator by a denominator.
Differential
\,\mathrm{d}xA differential marks the variable of integration or an infinitesimal change.
Second derivative
\frac{\mathrm{d}^2}{\mathrm{d}x^2}The second derivative measures how a rate of change itself changes.
Partial derivative operator
\frac{\partial}{\partial x}The partial derivative operator differentiates with respect to one variable while holding others fixed.
Double integral
\iintA double integral accumulates a function over a two-dimensional region.
Triple integral
\iiintA triple integral accumulates a function over a three-dimensional region.
Contour integral
\ointA contour integral integrates along a closed curve.
Laplacian
\nabla^2The Laplacian is a second-order differential operator related to curvature and diffusion.
Aleph null
\aleph_0Aleph null is the cardinality of countably infinite sets.
Continuum cardinality
\mathfrak{c}The continuum cardinality is the size of the real numbers.
Homomorphism arrow
\varphi:G\to HA homomorphism is a structure-preserving map between algebraic objects.
Normal subgroup
\triangleleftNormal subgroup notation identifies subgroups compatible with quotient group construction.
Tensor product
\otimesThe tensor product combines vector spaces, modules, or operators in multilinear algebra.
Direct sum
\oplusThe direct sum combines structures while keeping components independent.
Empty product
\prod_{i\in\varnothing}An empty product is conventionally equal to one.
Big O
O(n)Big O describes an upper bound on asymptotic growth.
Little o
o(n)Little o describes growth that becomes negligible compared with another function.
Big Theta
\Theta(n)Big Theta describes matching upper and lower asymptotic bounds.
Square
\squareA square symbol can mark a completed proof or act as a placeholder box.
Angle brackets
\langle x,y\rangleAngle brackets can denote vectors, inner products, or generated structures.
Norm
\|x\|A norm measures the size or length of a vector or object.
Evaluated at
\big|The vertical evaluation bar indicates substituting bounds after finding an antiderivative.
Capital gamma
\GammaCapital gamma often denotes the gamma function, a curve, or a group.
Capital lambda
\LambdaCapital lambda is used for operators, lattices, and parameters.
Capital sigma
\SigmaCapital sigma is used for sums, alphabets, covariance matrices, and surfaces.
Capital pi
\PiCapital pi is often used for products or projection operators.
Capital phi
\PhiCapital phi often names functions, maps, or the normal distribution CDF.
Kappa
\kappaKappa is commonly used for curvature, constants, and parameters.
Nu
\nuNu is often used for frequency, degrees of freedom, or measures.
Zeta
\zetaZeta appears in special functions and number theory.
Iota
\iotaIota can denote an inclusion map or a small indexed quantity.
Upsilon
\upsilonUpsilon is a Greek letter used as a variable or parameter.
Equals
=Equals states that two expressions have the same value.
Less than
<Less than compares two ordered quantities.
Greater than
>Greater than compares two ordered quantities.
Much less than
\llMuch less than indicates one quantity is significantly smaller than another.
Much greater than
\ggMuch greater than indicates one quantity is significantly larger than another.
Precedes
\precPrecedes denotes an ordering relation more general than ordinary less-than.
Succeeds
\succSucceeds denotes the reverse of a precedes relation.
Left arrow
\leftarrowA left arrow can show reverse implication, assignment, or direction.
Right arrow
\rightarrowA right arrow can show a map, implication, transition, or limit direction.
Long right arrow
\longrightarrowA long right arrow is used for emphasized maps or transitions.
Long if and only if
\LongleftrightarrowA long double arrow expresses equivalence or two-way implication.
Hook arrow
\hookrightarrowA hook arrow often denotes an inclusion or injective map.
Surjection arrow
\twoheadrightarrowA two-headed arrow often denotes a surjective map.
Bijection arrow
\leftrightarrowA two-way arrow can represent a one-to-one correspondence.
Converges to
a_n\to LConverges to describes a sequence or expression approaching a limit.
Divergence
\nabla\cdotDivergence measures net outward flow from a vector field.
Curl
\nabla\timesCurl measures local rotation of a vector field.
Surface integral
\iint_SA surface integral accumulates a quantity over a surface.
Line integral
\int_CA line integral accumulates a quantity along a curve.
Imaginary unit
iThe imaginary unit is defined by i squared equals negative one.
Real part
\operatorname{Re}The real part extracts the real component of a complex number.
Imaginary part
\operatorname{Im}The imaginary part extracts the coefficient of i in a complex number.
Complex conjugate
\overline{z}The complex conjugate changes the sign of the imaginary part.
Median
\operatorname{median}The median is the middle value of an ordered data set.
Correlation
\operatorname{corr}Correlation measures linear association between two variables.
Covariance
\operatorname{Cov}Covariance measures how two random variables vary together.
Sample mean
\bar{x}The sample mean is the average of observed values.
Estimator hat
\hat{\theta}A hat marks an estimated value of a parameter.
Approximately distributed as
\dot\simThis notation can indicate approximate distributional behavior.
Normal distribution
N(\mu,\sigma^2)The normal distribution is a bell-shaped probability distribution.
Permutation
P(n,k)A permutation counts ordered selections.
Combination
C(n,k)A combination counts unordered selections.
Minimum
\minMinimum returns the smallest value in a set or expression.
Maximum
\maxMaximum returns the largest value in a set or expression.
Argmin
\operatorname*{arg\,min}Argmin returns where a function reaches its minimum.
Argmax
\operatorname*{arg\,max}Argmax returns where a function reaches its maximum.
Logarithm
\logA logarithm is the inverse operation to exponentiation for a chosen base.
Common logarithm
\log_{10}The common logarithm uses base ten.
Hyperbolic sine
\sinhHyperbolic sine is a hyperbolic function defined using exponentials.
Hyperbolic cosine
\coshHyperbolic cosine is a hyperbolic function defined using exponentials.
Arcsine
\arcsinArcsine is the inverse function of sine on a restricted domain.
Arctangent
\arctanArctangent is the inverse function of tangent on a restricted domain.