The tables below set out the notation that must be used in A Level Mathematics examinations. Students will be expected to understand this notation without need for further explanation.

Set Notation 1

Set Notation

	Notation	Meaning
1.1	$\in$	is an element of
1.2	$\notin$	is not an element of
1.3	$\subseteq$	is a subset of
1.4	$\subset$	is a proper subset of
1.5	$\{x_1, x_2, \dots\}$	the set with elements $x_1, x_2, \dots$
1.6	$\{x : \dots\}$	the set of all $x$ such that $\dots$
1.7	$\text{n}(A)$	the number of elements in set $A$
1.8	$\varnothing$	the empty set
1.9	$\mathcal{E}$	the universal set
1.10	$A'$	the complement of the set $A$
1.11	$\mathbb{N}$	the set of natural numbers, $\{1, 2, 3, \dots\}$
1.12	$\mathbb{Z}$	the set of integers, $\{0, \pm 1, \pm 2, \pm 3, \dots\}$
1.13	$\mathbb{Z}^+$	the set of positive integers, $\{1, 2, 3, \dots\}$
1.14	$\mathbb{Z}_0^+$	the set of non-negative integers, $\{0, 1, 2, 3, \dots\}$
1.15	$\mathbb{R}$	the set of real numbers
1.16	$\mathbb{Q}$	the set of rational numbers, $\left\{\frac{p}{q} : p \in \mathbb{Z}, q \in \mathbb{Z}^+\right\}$
1.17	$\cup$	union
1.18	$\cap$	intersection
1.19	$(x, y)$	the ordered pair $x, y$
1.20	$[a, b]$	the closed interval $\{x \in \mathbb{R} : a \le x \le b\}$
1.21	$[a, b)$	the interval $\{x \in \mathbb{R} : a \le x < b\}$
1.22	$(a, b]$	the interval $\{x \in \mathbb{R} : a < x \le b\}$
1.23	$(a, b)$	the open interval $\{x \in \mathbb{R} : a < x < b\}$

Miscellaneous Symbols & Operations 2 & 3

Miscellaneous Symbols

	Notation	Meaning
2.1	$=$	is equal to
2.2	$\neq$	is not equal to
2.3	$\equiv$	is identical to or is congruent to
2.4	$\approx$	is approximately equal to
2.5	$\infty$	infinity
2.6	$\propto$	is proportional to
2.7	$\therefore$	therefore
2.8	$\because$	because
2.9	$<$	is less than
2.10	$\leqslant$ , $\le$	is less than or equal to, is not greater than
2.11	$>$	is greater than
2.12	$\geqslant$ , $\ge$	is greater than or equal to, is not less than
2.13	$p \Rightarrow q$	$p$ implies $q$ (if $p$ then $q$ )
2.14	$p \Leftarrow q$	$p$ is implied by $q$ (if $q$ then $p$ )
2.15	$p \iff q$	$p$ implies and is implied by $q$ ( $p$ is equivalent to $q$ )
2.16	$a$	first term for an arithmetic or geometric sequence
2.17	$l$	last term for an arithmetic sequence
2.18	$d$	common difference for an arithmetic sequence
2.19	$r$	common ratio for a geometric sequence
2.20	$S_n$	sum to $n$ terms of a sequence
2.21	$S_{\infty}$	sum to infinity of a sequence

Operations

	Notation	Meaning
3.1	$a + b$	$a$ plus $b$
3.2	$a - b$	$a$ minus $b$
3.3	$a \times b, ab, a \cdot b$	$a$ multiplied by $b$
3.4	$a \div b, \frac{a}{b}$	$a$ divided by $b$
3.5	$\sum_{i=1}^n a_i$	$a_1 + a_2 + \dots + a_n$
3.6	$\prod_{i=1}^n a_i$	$a_1 \times a_2 \times \dots \times a_n$
3.7	$\sqrt{a}$	the non-negative square root of $a$
3.8	$\|a\|$	the modulus of $a$
3.9	$n!$	$n$ factorial: $n! = n \times (n-1) \times \dots \times 2 \times 1, \ n \in \mathbb{N}; \ 0! = 1$
3.10	$\binom{n}{r}, \text{ }^{n}C_r, \text{ }_nC_r$	the binomial coefficient $\frac{n!}{r!(n-r)!}$ for $n, r \in \mathbb{Z}_0^+, \ r \le n$ or $\frac{n(n-1)\dots(n-r+1)}{r!}$ for $n \in \mathbb{Q}, \ r \in \mathbb{Z}_0^+$

Functions & Exponential/Logarithmic Functions 4 & 5

Functions

	Notation	Meaning
4.1	$f(x)$	the value of the function $f$ at $x$
4.2	$f : x \mapsto y$	the function $f$ maps the element $x$ to the element $y$
4.3	$f^{-1}$	the inverse function of the function $f$
4.4	$gf$	the composite function of $f$ and $g$ which is defined by $gf(x) = g(f(x))$
4.5	$\lim_{x \to a} f(x)$	the limit of $f(x)$ as $x$ tends to $a$
4.6	$\Delta x, \delta x$	an increment of $x$
4.7	$\frac{dy}{dx}$	the derivative of $y$ with respect to $x$
4.8	$\frac{d^n y}{dx^n}$	the $n$ th derivative of $y$ with respect to $x$
4.9	$f'(x), f''(x), \dots, f^{(n)}(x)$	the first, second, \dots, $n$ th derivatives of $f(x)$ with respect to $x$
4.10	$\dot{x}, \ddot{x}, \dots$	the first, second, \dots derivatives of $x$ with respect to $t$
4.11	$\int y\,dx$	the indefinite integral of $y$ with respect to $x$
4.12	$\int_{a}^{b} y\,dx$	the definite integral of $y$ with respect to $x$ between the limits $x = a$ and $x = b$

Exponential and Logarithmic Functions

	Notation	Meaning
5.1	$e$	base of natural logarithms
5.2	$e^x, \exp x$	exponential function of $x$
5.3	$\log_a x$	logarithm to the base $a$ of $x$
5.4	$\ln x, \log_e x$	natural logarithm of $x$

Trigonometric Functions & Vectors 6 & 7

Trigonometric Functions

	Notation	Meaning
6.1	$\sin, \cos, \tan$ , $\csc, \sec, \cot$	the trigonometric functions
6.2	$\sin^{-1}, \cos^{-1}, \tan^{-1}$ , $\arcsin, \arccos, \arctan$	the inverse trigonometric functions
6.3	$^{\circ}$	degrees
6.4	$\text{rad}$	radians

Vectors

	Notation	Meaning
7.1	$\mathbf{a}, \underline{a}, \vec{a}$	the vector $\mathbf{a}$ ; these alternatives apply throughout section 9 (vectors)
7.2	$\overrightarrow{AB}$	the vector represented in magnitude and direction by the directed line segment $AB$
7.3	$\mathbf{\hat{a}}$	a unit vector in the direction of $\mathbf{a}$
7.4	$\mathbf{i}, \mathbf{j}, \mathbf{k}$	unit vectors in the directions of the cartesian coordinate axes
7.5	$\|\mathbf{a}\|, a$	the magnitude of $\mathbf{a}$
7.6	$\|\overrightarrow{AB}\|, AB$	the magnitude of $\overrightarrow{AB}$
7.7	$\begin{pmatrix} a \\ b \end{pmatrix}, a\mathbf{i} + b\mathbf{j}$	column vector and corresponding unit vector notation
7.8	$\mathbf{r}$	position vector
7.9	$\mathbf{s}$	displacement vector
7.10	$\mathbf{v}$	velocity vector
7.11	$\mathbf{a}$	acceleration vector

Probability & Statistics 8

Probability and Statistics Symbols

	Notation	Meaning
8.1	$A, B, C$ , etc.	events
8.2	$A \cup B$	union of the events $A$ and $B$
8.3	$A \cap B$	intersection of the events $A$ and $B$
8.4	$P(A)$	probability of the event $A$
8.5	$A'$	complement of the event $A$
8.6	$P(A\|B)$	probability of the event $A$ conditional on the event $B$
8.7	$X, Y, R$ , etc.	random variables
8.8	$x, y, r$ , etc.	values of the random variables $X, Y, R$ , etc.
8.9	$x_1, x_2, \dots$	observations
8.10	$f_1, f_2, \dots$	frequencies with which the observations $x_1, x_2, \dots$ occur
8.11	$p(x), P(X = x)$	probability function of the discrete random variable $X$
8.12	$p_1, p_2, \dots$	probabilities of the values $x_1, x_2, \dots$ of the discrete random variable $X$
8.13	$E(X)$	expectation of the random variable $X$
8.14	$Var(X)$	variance of the random variable $X$
8.15	$\sim$	has the distribution
8.16	$\text{B}(n, p)$	binomial distribution with parameters $n$ and $p$ , where $n$ is the number of trials and $p$ is the probability of success in a trial
8.17	$q$	$q = 1 - p$ for binomial distribution
8.18	$\text{N}(\mu, \sigma^2)$	Normal distribution with mean $\mu$ and variance $\sigma^2$
8.19	$Z \sim \text{N}(0, 1)$	standard Normal distribution
8.20	$\phi$	probability density function of the standardised Normal variable with distribution $\text{N}(0, 1)$
8.21	$\Phi$	corresponding cumulative distribution function
8.22	$\mu$	population mean
8.23	$\sigma^2$	population variance
8.24	$\sigma$	population standard deviation
8.25	$\bar{x}$	sample mean
8.26	$s^2$	sample variance
8.27	$s$	sample standard deviation
8.28	$\text{H}_0$	Null hypothesis
8.29	$\text{H}_1$	Alternative hypothesis
8.30	$r$	product moment correlation coefficient for a sample
8.31	$\rho$	product moment correlation coefficient for a population

Mechanics Notation 9

Mechanics Symbols

	Notation	Meaning
9.1	$\text{kg}$	kilograms
9.2	$\text{m}$	metres
9.3	$\text{km}$	kilometres
9.4	$\text{m/s}, \text{m}\ \text{s}^{-1}$	metres per second (velocity)
9.5	$\text{m/s}^2, \text{m}\ \text{s}^{-2}$	metres per second per second (acceleration)
9.6	$F$	Force or resultant force
9.7	$\text{N}$	Newton
9.8	$\text{N}\ \text{m}$	Newton metre (moment of a force)
9.9	$t$	time
9.10	$s$	displacement
9.11	$u$	initial velocity
9.12	$v$	velocity or final velocity
9.13	$a$	acceleration
9.14	$g$	acceleration due to gravity
9.15	$\mu$	coefficient of friction

Mathematical Notations - A-Levels Maths

Set Notation 1

Set Notation

Miscellaneous Symbols & Operations 2 & 3

Miscellaneous Symbols

Operations

Functions & Exponential/Logarithmic Functions 4 & 5

Functions

Exponential and Logarithmic Functions

Trigonometric Functions & Vectors 6 & 7

Trigonometric Functions

Vectors

Probability & Statistics 8

Probability and Statistics Symbols

Mechanics Notation 9

Mechanics Symbols