Terms and Symbols Used in AI

Mathematical notation lets us describe data, models and operations precisely. This page is a reference for the symbols used throughout the guide. You do not need to memorise everything at once; return here whenever unfamiliar notation appears.

Letter styles and decorations

The style of a letter often tells you what kind of object it represents.

Notation	Meaning
$x$	Usually a scalar, meaning one numerical value.
$\mathbf{x}$	Usually a vector, meaning an ordered list of values.
$X$	Usually a matrix or a collection of data.
$\mathcal{X}$	Usually a set or a space of possible values.
$\theta$	Commonly a parameter or a collection of model parameters.
$i,j,k$	Commonly indices used to identify positions or items.
$m,n,d,p$	Commonly sizes, counts or dimensions.
$\hat{x}$	An estimated or predicted value of $x$ .
$\bar{x}$	The average of a collection of values named $x$ .
$\tilde{x}$	A transformed, adjusted or approximate version of $x$ .
$x^*$	An optimal, selected or otherwise distinguished value of $x$ .
$x'$	A modified or alternative version of $x$ .
$x^{(i)}$	The $i$ -th observation or item.
$x_i$	The $i$ -th component of $x$ .
$x_{ij}$	The entry at row $i$ and column $j$ .
$x^2$	$x$ raised to the power $2$ .
$X^\top$	The transpose of $X$ .
$X^{-1}$	The inverse of $X$ , when it exists.

Common AI and data notation

These conventions appear frequently in machine learning, although an author may choose different letters.

Notation	Meaning
$X$	The input data or feature matrix.
$\mathbf{x}$	One input example or feature vector.
$x_j$	The $j$ -th feature of an input.
$\mathbf{x}^{(i)}$	The $i$ -th input example.
$x_j^{(i)}$	The $j$ -th feature of the $i$ -th example.
$\mathbf{y}$	A collection of target or output values.
$y^{(i)}$	The target associated with the $i$ -th example.
$\hat{y}$	A predicted output.
$\hat{\mathbf{y}}$	A collection of predicted outputs.
$\mathcal{D}$	A dataset.
$n$ or $m$	The number of examples, depending on the convention being used.
$d$ or $p$	The number of features or dimensions.
$k$ or $c$	The number of classes or groups.
$\mathbf{w}$	A vector of weights.
$b$	A bias or intercept.
$\theta$	The parameters of a model.
$f$	A function or model.
$f_\theta$	A function or model controlled by parameters $\theta$ .
$\mathcal{L}$	A loss or objective function.
$\lambda$	A coefficient or weighting value whose exact meaning depends on context.

Number systems and special values

Notation	Meaning
$\mathbb{N}$	The set of natural numbers.
$\mathbb{Z}$	The set of integers.
$\mathbb{Q}$	The set of rational numbers.
$\mathbb{R}$	The set of real numbers.
$\mathbb{C}$	The set of complex numbers.
$\mathbb{R}^d$	The set of real-valued vectors with $d$ components.
$\mathbb{R}^{m\times n}$	The set of real-valued matrices with $m$ rows and $n$ columns.
$0$	The number zero.
$\mathbf{0}$	A vector or matrix containing only zeros.
$1$	The number one.
$\mathbf{1}$	A vector or matrix containing only ones.
$\infty$	Infinity or an unbounded quantity.
$\pi$	The mathematical constant pi.
$e$	The base of the natural logarithm.

Some sources include $0$ in $\mathbb{N}$ and others begin at $1$ , so the intended definition should be stated when it matters.

Equality, comparison and approximation

Notation	Meaning
$a=b$	$a$ is equal to $b$ .
$a\neq b$	$a$ is not equal to $b$ .
$a\approx b$	$a$ is approximately equal to $b$ .
$a\equiv b$	$a$ and $b$ are identical, equivalent or equal by definition, depending on context.
$a:=b$	$a$ is defined to be $b$ .
$a\propto b$	$a$ is proportional to $b$ .
$a<b$	$a$ is less than $b$ .
$a>b$	$a$ is greater than $b$ .
$a\leq b$	$a$ is less than or equal to $b$ .
$a\geq b$	$a$ is greater than or equal to $b$ .
$a\ll b$	$a$ is much smaller than $b$ .
$a\gg b$	$a$ is much larger than $b$ .

Basic arithmetic

Notation	Meaning
$a+b$	The sum of $a$ and $b$ .
$a-b$	The difference between $a$ and $b$ .
$ab$ , $a\cdot b$ or $a\times b$	The product of $a$ and $b$ .
$\frac{a}{b}$	$a$ divided by $b$ .
$a^b$	$a$ raised to the power $b$ .
$\sqrt{x}$	The square root of $x$ .
$\sqrt[n]{x}$	The $n$ -th root of $x$ .
$\lvert x\rvert$	The absolute value of the number $x$ .
$a\bmod b$	The remainder after dividing $a$ by $b$ .
$\lfloor x\rfloor$	$x$ rounded down to the nearest integer.
$\lceil x\rceil$	$x$ rounded up to the nearest integer.
$\pm$	Plus or minus.
$\frac{1}{x}$	The reciprocal of $x$ .

Sets and intervals

Notation	Meaning
$A$	A set.
$\{a,b,c\}$	A set containing the listed elements.
$\{x:P(x)\}$	The set of values $x$ for which condition $P(x)$ is true.
$x\in A$	$x$ belongs to set $A$ .
$x\notin A$	$x$ does not belong to set $A$ .
$A\subseteq B$	Every element of $A$ also belongs to $B$ .
$A\subset B$	$A$ is a proper subset of $B$ .
$A\supseteq B$	$A$ contains every element of $B$ .
$A\cup B$	The union of $A$ and $B$ .
$A\cap B$	The intersection of $A$ and $B$ .
$A\setminus B$	The elements in $A$ that are not in $B$ .
$A^c$	The complement of $A$ .
$\varnothing$	The empty set.
$\lvert A\rvert$	The number of elements in set $A$ .
$A\times B$	The Cartesian product of $A$ and $B$ .
$[a,b]$	The interval from $a$ to $b$ containing both endpoints.
$(a,b)$	The interval from $a$ to $b$ excluding both endpoints.
$[a,b)$	The interval containing $a$ but excluding $b$ .
$(a,b]$	The interval excluding $a$ but containing $b$ .

Logic

Notation	Meaning
$\neg P$	Statement $P$ is false.
$P\land Q$	Both $P$ and $Q$ are true.
$P\lor Q$	At least one of $P$ or $Q$ is true.
$P\Rightarrow Q$	$P$ implies $Q$ .
$P\Leftrightarrow Q$	$P$ and $Q$ are logically equivalent.
$\forall x$	For every value of $x$ .
$\exists x$	There exists at least one value of $x$ .
$\nexists x$	No such value of $x$ exists.
$\mathbb{1}[P]$	The indicator that equals $1$ when $P$ is true and $0$ otherwise.

Functions and mappings

Notation	Meaning
$f(x)$	The output of function $f$ for input $x$ .
$f:A\rightarrow B$	Function $f$ maps values from set $A$ to set $B$ .
$x\mapsto f(x)$	Input $x$ is mapped to output $f(x)$ .
$f\circ g$	The composition that applies $g$ and then applies $f$ .
$f^{-1}$	The inverse of function $f$ , when it exists.
$\operatorname{dom}(f)$	The domain of $f$ , containing its allowed inputs.
$\operatorname{range}(f)$	The range of $f$ , containing the outputs it produces.
$\log(x)$	A logarithm whose base depends on context.
$\ln(x)$	The natural logarithm of $x$ .
$\log_b(x)$	The logarithm of $x$ to base $b$ .
$\exp(x)$	The exponential function $e^x$ .

The notation $f^{-1}$ means an inverse function, while $x^{-1}$ usually means the reciprocal of a number or the inverse of a matrix.

Indices, sequences and collections

Notation	Meaning
$x_i$	The item at index $i$ .
$x_{i:j}$	A range or slice of items from index $i$ to index $j$ , with endpoint rules depending on context.
$(x_1,\ldots,x_n)$	An ordered sequence of $n$ values.
$\{x_i\}_{i=1}^{n}$	A collection indexed from $1$ through $n$ .
$i=1,\ldots,n$	Index $i$ takes the integer values from $1$ through $n$ .
$A_{ij}$	The entry in row $i$ and column $j$ of matrix $A$ .
$A_{i:}$	The $i$ -th row of $A$ .
$A_{:j}$	The $j$ -th column of $A$ .
$\lvert\mathbf{x}\rvert$	The number of items in a sequence when used in that context.
$\ldots$	Values have been omitted because the pattern is understood.

Sums, products and reductions

Notation	Meaning
$\sum_{i=1}^{n}x_i$	The sum of the values $x_1$ through $x_n$ .
$\prod_{i=1}^{n}x_i$	The product of the values $x_1$ through $x_n$ .
$\min(x_1,\ldots,x_n)$	The smallest of the listed values.
$\max(x_1,\ldots,x_n)$	The largest of the listed values.
$\min_{x\in S}f(x)$	The smallest value produced by $f(x)$ over $S$ .
$\max_{x\in S}f(x)$	The largest value produced by $f(x)$ over $S$ .
$\arg\min_{x\in S}f(x)$	The value or values of $x$ at which $f(x)$ is smallest.
$\arg\max_{x\in S}f(x)$	The value or values of $x$ at which $f(x)$ is largest.
$\operatorname{mean}(x)$	The arithmetic average of the values in $x$ .

Vectors, matrices and tensors

Notation	Meaning
$\mathbf{x}$	A vector.
$X$	A matrix.
$\mathcal{T}$	A tensor, when that convention is used.
$\mathbf{x}\in\mathbb{R}^d$	$\mathbf{x}$ is a real-valued vector with $d$ components.
$X\in\mathbb{R}^{m\times n}$	$X$ is a real-valued matrix with $m$ rows and $n$ columns.
$X^\top$	The transpose of $X$ .
$X^{-1}$	The inverse of $X$ , when it exists.
$X^+$	The pseudoinverse of $X$ .
$I$	An identity matrix.
$\mathbf{0}$	A vector or matrix containing only zeros.
$\mathbf{1}$	A vector or matrix containing only ones.
$AB$	The matrix product of $A$ and $B$ .
$A\odot B$	The elementwise product of $A$ and $B$ .
$\mathbf{x}^\top\mathbf{y}$	The dot product of vectors $\mathbf{x}$ and $\mathbf{y}$ .
$\|\mathbf{x}\|$	A norm measuring the size of vector $\mathbf{x}$ .
$\|\mathbf{x}\|_1$	The sum of the absolute values of the components of $\mathbf{x}$ .
$\|\mathbf{x}\|_2$	The Euclidean length of $\mathbf{x}$ .
$\|X\|_F$	The Frobenius norm of matrix $X$ .
$\det(X)$	The determinant of square matrix $X$ .
$\operatorname{tr}(X)$	The sum of the diagonal entries of square matrix $X$ .
$\operatorname{rank}(X)$	The number of linearly independent rows or columns of $X$ .
$X_{i,:}$	The $i$ -th row of $X$ .
$X_{:,j}$	The $j$ -th column of $X$ .
$\operatorname{shape}(X)$	The dimensions of $X$ .

Calculus notation

This section introduces the symbols for recognition. The Calculus module explains the ideas behind them.

Notation	Meaning
$\frac{dy}{dx}$	The derivative of $y$ with respect to $x$ .
$f'(x)$	The derivative of $f$ at $x$ .
$\frac{\partial f}{\partial x}$	The partial derivative of $f$ with respect to $x$ .
$\nabla f$	The gradient of $f$ .
$\nabla_{\mathbf{x}}f$	The gradient of $f$ with respect to vector $\mathbf{x}$ .
$\int f(x)\,dx$	The indefinite integral of $f(x)$ .
$\int_a^b f(x)\,dx$	The definite integral of $f(x)$ from $a$ to $b$ .
$\lim_{x\to a}f(x)$	The limit of $f(x)$ as $x$ approaches $a$ .
$\Delta x$	A finite change in $x$ .

Probability notation

Notation	Meaning
$X$	A random variable when used in a probability context.
$x$	A possible observed value of random variable $X$ .
$P(A)$	The probability of event $A$ .
$P(X=x)$	The probability that $X$ takes the value $x$ .
$P(A\mid B)$	The probability of $A$ given that $B$ has occurred.
$P(A,B)$	The joint probability of $A$ and $B$ .
$p(x)$	A probability mass or probability density value associated with $x$ .
$p(x\mid y)$	A conditional probability mass or density.
$X\sim P$	Random variable $X$ follows distribution $P$ .
$X\perp Y$	Random variables $X$ and $Y$ are independent.
$\mathbb{E}[X]$	The expected value of $X$ .
$\operatorname{Var}(X)$	The variance of $X$ .
$\operatorname{Cov}(X,Y)$	The covariance of $X$ and $Y$ .
$\operatorname{Corr}(X,Y)$	The correlation between $X$ and $Y$ .

Statistics and estimation notation

Notation	Meaning
$\bar{x}$	The sample mean.
$\mu$	Commonly a population mean.
$\sigma$	Commonly a population standard deviation.
$\sigma^2$	Commonly a population variance.
$s$	Commonly a sample standard deviation.
$s^2$	Commonly a sample variance.
$n$	The number of observations.
$\hat{\theta}$	An estimate of parameter $\theta$ .
$\tilde{x}$	A transformed, adjusted or approximate version of $x$ .

Optimisation notation

Notation	Meaning
$f(x)$	A function being evaluated or optimised.
$\mathcal{L}(\theta)$	A loss or objective that depends on parameters $\theta$ .
$\theta^*$	An optimal value of $\theta$ .
$\min_\theta\mathcal{L}(\theta)$	The smallest loss obtainable by varying $\theta$ .
$\arg\min_\theta\mathcal{L}(\theta)$	The parameter value or values that produce the smallest loss.
$\max_\theta f(\theta)$	The largest value obtainable by varying $\theta$ .
$\arg\max_\theta f(\theta)$	The parameter value or values that produce the largest value.
$\eta$	Commonly a step size or learning rate.
$\theta_t$	The value of $\theta$ at step $t$ .

Common mathematical and AI terms

Term	Meaning
Scalar	A single numerical value.
Variable	A symbol whose value may change or be unknown.
Constant	A value treated as fixed in the current context.
Coefficient	A value that multiplies another quantity.
Parameter	A value that controls a model or function.
Estimate	A value calculated as an approximation of an unknown quantity.
Prediction	An output produced by a model for given input data.
Feature	An input property or measurement used by a model.
Target	The value a supervised model is intended to predict.
Label	A target value, especially a category in classification.
Observation	One recorded item or example in a dataset.
Sample	One observation or a collection drawn from a larger population, depending on context.
Dimension	The number of components or independent directions in an object or space.
Shape	The sizes of an array along each of its axes.
Index	A value used to identify a position or item.
Axis	A direction along which the entries of an array are arranged.
Vector	An ordered one-dimensional collection of values.
Matrix	A rectangular two-dimensional arrangement of values.
Tensor	A multidimensional array of values.
Function	A rule that assigns an output to each allowed input.
Input	A value supplied to a function or model.
Output	A value produced by a function or model.
Domain	The set of inputs for which a function is defined.
Range	The set of outputs a function actually produces.
Constraint	A condition that an allowed value or solution must satisfy.
Objective	A quantity that an optimisation process tries to minimise or maximise.
Loss	A numerical measure of how wrong a prediction or model is.
Norm	A function that measures the size or length of a vector or matrix.
Distance	A measure of how far apart two objects are.
Distribution	A description of how probability is assigned across possible values.
Event	A set of possible outcomes in a probability setting.
Random variable	A variable whose value is determined by a random outcome.
Expected value	The probability-weighted average value of a random variable.
Variance	A measure of how widely values vary around their mean.
Gradient	A vector containing the partial derivatives of a scalar-valued function.
Derivative	A measure of how a quantity changes as another quantity changes.

You are not expected to memorise this reference before continuing. Learn the notation as it appears, and return to this page whenever a symbol is unfamiliar.

2.3.1 Terms and Symbols Used in AI