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1 Set Notation

Ref.	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	$\{x_1, x_2, ...\}$	the set with elements $x_1, x_2, ...$
1.6	$\{x: ...\}$	the set of all $x$ such that ...
1.7	$n(A)$	the number of elements in set $A$
1.8	$\varnothing$	the empty set
1.9	$\varepsilon$	the universal set
1.10	$A'$	the complement of the set $A$
1.11	$\mathbb{N}$	the set of natural numbers, $\{1, 2, 3, ...\}$
1.12	$\mathbb{Z}$	the set of integers, $\{0, \pm1, \pm2, \pm3, ...\}$
1.13	$\mathbb{Z}^{+}$	the set of positive integers, $\{1, 2, 3, ...\}$
1.14	$\mathbb{Z}_0^{+}$	the set of non-negative integers, $\{0, 1, 2, 3, ...\}$
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 $\{x \in \mathbb{R}: a < x < b\}$

2 Miscellaneous Symbols

Ref.	Notation	Meaning
2.1	$=$	is equal to
2.2	$\ne$	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	$\le, \leqslant$	is less than or equal to, is not greater than
2.11	$>$	is greater than
2.12	$\ge, \geqslant$	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	$p \Leftrightarrow q$	$p$ 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

3 Operations

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

4 Functions

Ref.	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	$f'(x)$ , $f''(x)$ , ..., $f^{(n)}(x)$	the first, second, ..., $n$ th derivatives of $f(x)$ with respect to $x$
4.10	$\dot{x}$ , $\ddot{x}$ , ...	the first, second, ... 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 $x = b$

5 Exponential and Logarithmic Functions

Ref.	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 $x$

6 Trigonometric Functions

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

9 Vectors

Ref.	Notation	Meaning
9.1	$\mathbf{a}$ , $\underline{a}$ , $\vec{a}$	the vector $\mathbf{a}$ , $\underline{a}$ , $\vec{a}$ ; these alternatives apply throughout section 9
9.2	$\overrightarrow{AB}$	the vector represented in magnitude and direction by the directed line segment $AB$
9.3	$\hat{\mathbf{a}}$	a unit vector in the direction of $\mathbf{a}$
9.4	$\mathbf{i}$ , $\mathbf{j}$ , $\mathbf{k}$	unit vectors in the directions of the cartesian coordinate axes
9.5	$\|\mathbf{a}\|$ , $a$	the magnitude of $\mathbf{a}$
9.6	$\|\overrightarrow{AB}\|$ , $AB$	the magnitude of $\overrightarrow{AB}$
9.7	$\begin{pmatrix}a\\b\end{pmatrix}$ , $ai + bj$	column vector and corresponding unit vector notation
9.8	$\mathbf{r}$	position vector
9.9	$\mathbf{s}$	displacement vector
9.10	$\mathbf{v}$	velocity vector
9.11	$\mathbf{a}$	acceleration vector

11 Probability and Statistics

Ref.	Notation	Meaning
11.1	$A$ , $B$ , $C$ , etc.	events
11.2	$A \cup B$	union of the events $A$ and $B$
11.3	$A \cap B$	intersection of the events $A$ and $B$
11.4	$\mathrm{P}(A)$	probability of the event $A$
11.5	$A'$	complement of the event $A$
11.6	$\mathrm{P}(A\|B)$	probability of the event $A$ conditional on the event $B$
11.7	$X$ , $Y$ , $R$ , etc.	random variables
11.8	$x$ , $y$ , $r$ , etc.	values of the random variables $X$ , $Y$ , $R$ etc.
11.9	$x_1$ , $x_2$ , ...	values of observations
11.10	$f_1$ , $f_2$ , ...	frequencies with which the observations $x_1$ , $x_2$ , ... occur
11.11	$\mathrm{p}(x)$ , $\mathrm{P}(X = x)$	probability function of the discrete random variable $X$
11.12	$p_1$ , $p_2$ , ...	probabilities of the values $x_1$ , $x_2$ , ... of the discrete random variable $X$
11.13	$\mathrm{E}(X)$	expectation of the random variable $X$
11.14	$\mathrm{Var}(X)$	variance of the random variable $X$
11.15	$\sim$	has the distribution
11.16	$\mathrm{B}(n, p)$	binomial distribution with parameters $n$ and $p$ , where $n$ is the number of trials and $p$ is the probability of success in a trial
11.17	$q$	$q = 1 - p$ for binomial distribution
11.18	$\mathrm{N}(\mu, \sigma^2)$	Normal distribution with mean $\mu$ and variance $\sigma^2$
11.19	$Z \sim \mathrm{N}(0, 1)$	standard Normal distribution
11.20	$\phi$	probability density function of the standardised Normal variable with distribution $\mathrm{N}(0, 1)$
11.21	$\Phi$	corresponding cumulative distribution function
11.22	$\mu$	population mean
11.23	$\sigma^2$	population variance
11.24	$\sigma$	population standard deviation
11.25	$\bar{x}$	sample mean
11.26	$s^2$	sample variance
11.27	$s$	sample standard deviation
11.28	$H_0$	Null hypothesis
11.29	$H_1$	Alternative hypothesis
11.30	$r$	product moment correlation coefficient for a sample
11.31	$\rho$	product moment correlation coefficient for a population

12 Mechanics

Ref.	Notation	Meaning
12.1	kg	kilograms
12.2	m	metres
12.3	km	kilometres
12.4	m/s, $ms^{-1}$	metres per second (velocity)
12.5	m/s², $ms^{-2}$	metres per second per second (acceleration)
12.6	$F$	Force or resultant force
12.7	N	Newton
12.8	N m	Newton metre (moment of a force)
12.9	$t$	time
12.10	$s$	displacement
12.11	$u$	initial velocity
12.12	$v$	velocity or final velocity
12.13	$a$	acceleration
12.14	$g$	acceleration due to gravity
12.15	$\mu$	coefficient of friction

Mathematical Notation - A-Levels Maths

5c. Mathematical notation

1 Set Notation

2 Miscellaneous Symbols

3 Operations

4 Functions

5 Exponential and Logarithmic Functions

6 Trigonometric Functions

9 Vectors

11 Probability and Statistics

12 Mechanics