Mathematical Notation - a MaS Tutor.

Set notation:
1. $\in$ is an element of
2. $\notin$ is not an element of
3. $\subset$ is a proper subset of
4. $\subseteq$ is a subset of
5. ${x_{1}, x_{2}, x_{3}, ...}$ the set with elements $x_{1}$ , $x_{2}$ , $x_{3}$ , ...
6. ${x : ...}$ the set of all $x$ such that ...
7. $n (A)$ the number of elements in the set $A$ , it's size
8. ∅ the empty set
9. ℰ the universal set
10. $A^{'}$ the complement of the set $A$
11. $ℕ$ (or $ℤ^{+}$ ) the set of natural numbers, ${1, 2, 3, ...}$
12. $ℤ$ the set of integers, ${..., -3, -2, -1, 0, +1, +2, +3, ...}$
13. $ℤ_{0}^{+}$ the set of non-negative integers, ${0,1,2,3,...}$
14. $ℝ$ the set of real numbers
15. $ℚ$ the set of rational numbers, ${p / q : p \in ℤ, q \in ℕ}$
16. $ℂ$ the set of complex numbers, $x + y i$
17. $\cup$ union
18. $\cap$ intersection
19. $(x, y)$ the ordered pair $x$ , $y$
20. $[a, b]$ the closed interval ${x \in ℝ : a \leq x \leq b}$
21. $[a, b)$ the interval ${x \in ℝ : a \leq x < b}$
22. (a, b] the interval {x ∈ ℝ: a < x ≤ b}
23. $(a, b)$ the open interval ${x \in ℝ : a < x < b}$
Relationships:
1. $=$ is equal to
2. $\neq$ is not equal to
3. $\equiv$ is identical to or is congruent to
4. $\approx$ is approximately equal to
5. ∞ infinity
6. ∝ is proportional to
7. ∴ therefore
8. ∵ because
9. < is less than
10. ≤ is less than or equal to, is not greater than
11. > is greater than
12. ≥ is greater than or equal to, is not less than
13. $p \Rightarrow q$ $p$ implies $q$ (if $p$ then $q$ )
14. $p \Leftarrow q$ $p$ is implied by $q$ (if $q$ then $p$ )
15. $p \Leftrightarrow q$ $p$ implies and is implied by $q$ ( $p$ is equivalent to $q$ )
16. $p ⟼ q$ $p$ maps to $q$ .
Sequences:
1. $a$ , first term for an arithmetic or geometric sequence
2. $l$ , last term for an arithmetic sequence
3. $d$ , common difference for an arithmetic sequence
4. $r$ , common ratio for a geometric sequence
5. $S_{n}$ , sum to $n$ terms of a sequence
6. $S_{\infty}$ , sum to infinity of a sequence
Operations:
1. $a + b$ , $a$ plus $b$
2. $a - b$ , $a$ minus $b$
3. $a \times b$ , $a b$ , $a \cdot b$ , $a$ multiplied by $b$
4. $a \div b$ , $a / b$ , $\frac{a}{b}$ , $a$ divided by $b$
5. $Σ_{i = m}^{n} a_{i} = a_{m} + a_{m + 1} + a_{m + 2} + ... + a_{n}$
6. $Π_{i = m}^{n} a_{i} = a_{m} \times a_{m + 1} \times a_{m + 2} \times ... \times a_{n}$
7. $\sqrt{a}$ the non-negative square root of $a$
8. $| a |$ the modulus of $a$
9. $n!$ $n$ factorial
10. $(\begin{matrix} n \\ r \end{matrix}) = {}^{n}C_{r} = {}_{n}C_{r}$ the binomial coefficient $\frac{n!}{r! (n - r)!}$ for $n$ , $r$ $\in ℤ_{0}^{+}$ $n \leq r$ , or $\frac{n (n - 1) (n - 2) \times ... \times (n - r + 1)}{r!}$ $n\inℚ$ , $r \in ℤ_{0}^{+}$ .
Functions:
1. $f (x)$ , the value of the function f at x
2. $f : x ⟼ y$ , the function f maps the element x to the element y
3. $f^{-1} (x)$ , the inverse function of the function $f (x)$
4. $g f$ , the composite function of $f$ and $g$ which is defined by $g f (x) = g (f (x))$
5. $\lim_{x \to a} f (x)$ , the limit of $f (x)$ as $x$ tends to $a$
6. $δ x$ , $Δ x$ , an increment of $x$
7. $\frac{d y}{d x}$ , the derivative of $y$ with respect to $x$
8. $\frac{d^{n} y}{{d x}^{n}}$ , the $n^{th}$ derivative of $y$ with respect to $x$
9. $f^{'} (x)$ , $f^{''} (x)$ , $f^{(n)}$ , the first, second, ..., $n^{th}$ derivatives of $f (x)$ with respect to $x$
10. $\dot{x}$ , $\overset{..}{x}$ , ..., the first, second, ... derivatives of $x$ with respect to $t$
11. $\int y d x$ , the indefinite integral of $y$ with respect to $x$
12. $\int_{a}^{b} y d x$ , the definite integral of $y$ with respect to $x$ between the limits $x = a$ and $x = b$
Exponential and Logarithmic Functions
1. $e$ base of natural logarithms
2. $e^{x}$ , $exp x$ exponential function of $x$
3. ${log}_{a} x$ logarithm to the base $a$ of $x$
4. $lg x$ logarithm to the base 10 of $x$
5. $ln x$ , ${log}_{e} x$ natural logarithm of $x$
Trigonometric Functions
1. sin θ, cos θ, tan θ, cosec θ, sec θ, cot θ, the trigonometric functions
2. ${sin}^{-1} θ = arcsin θ$ , ${cos}^{-1} θ = arccos θ$ , ${tan}^{-1} θ = arctan θ$ , the inverse trigonometric functions
3. °, degrees
4. rad, radians
Vectors:
1. $a$ , $\underset{~}{a}$ , first denotes a vector in typeset, second in handwriting
2. $\overset{⇀}{A B}$ , the vector represented in magnitude and direction by the directed line segment AB
3. $\hat{a}$ , a unit vector in the direction of $a$
4. i, j, k, unit vectors in the directions of the cartesian coordinate axes
5. $| a |$ , $a$ , the magnitude of a
6. $| A B |$ , $A B$ , the magnitude of $A B$
7. $(\begin{matrix} a \\ b \end{matrix})$ , $a i + b j$ , i j column vector and corresponding unit vector notation
8. $r$ , position vector
9. $s$ , displacement vector
10. $v$ , velocity vector
11. $a$ , acceleration vector
Probability and Statistics:
1. $A$ , $B$ , $C$ , etc. different events
2. $A \cup B$ union of the events, $A$ and $B$
3. A ∩ B, intersection of the events A and B
4. $P (A)$ , probability of the event $A$
5. A′ complement of the event A
6. $P (A | B)$ probability of the event $A$ conditional on the event $B$
7. X, Y, R, etc., random variables
8. x, y, r, etc., values of the random variables X, Y, R etc.
9. x₁, x₂, etc., observations
10. f₁, f₂, etc., frequencies with which the observations x₁, x₂, etc., occur
11. p(x), P(X = x), probability function of the discrete random variable X
12. p₁, p₂, etc., probabilities of the values x₁, x₂, etc., of the discrete random variable X
13. $E (X)$ , expectation of the random variable $X$
14. Var(X), variance of the random variable X
15. ~, has the distribution
16. $B (n, p)$ binomial distribution with parameters $n$ , the number of trials, and $p$ , the probability of one outcome in a trial, and $q (= 1 - p)$ is the probability of the other outcome;
17. N(μ, σ²), Normal distribution with mean μ and variance σ²
18. Z ~ N(0,1), standard Normal distribution
19. φ, probability density function of the standardised Normal variable with distribution N(0, 1)
20. Φ, corresponding cumulative distribution function
21. $µ$ , population mean
22. $var (x)$ , $σ^{2}$ , population variance
23. $σ$ , population standard deviation
24. $\overline{x}$ , sample mean
25. s², sample variance
26. s, sample standard deviation
27. $H_{0}$ , Null hypothesis
28. $H_{1}$ , Alternative hypothesis
29. $r$ , product moment correlation coefficient for a sample
30. $ρ$ , product moment correlation coefficient for a population
Mechanics:
1. $k g$ , kilograms
2. $m$ , metres
3. $k m$ , kilometres
4. $m / s$ , $m s^{-1}$ , metres per second (speed or velocity);
5. $m / s^{2}$ , $m s^{-2}$ , metres per second per second (acceleration);
6. $F$ force or resultant force
7. $N$ , Newton
8. $N m$ , Newton metre (moment of a force)
9. $t$ , time
10. $s$ , displacement
11. $u$ , initial velocity
12. $v$ , speed, velocity or final velocity
13. $a$ , acceleration
14. $g$ , acceleration due to gravity
15. $µ$ , coefficient of friction.

References:: Guidance taken from qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/a-level-l3-mathematics-specification-issue4.pdf;; Symbol code from www.toptal.com/designers/htmlarrows/math/; mathML from https://developer.mozilla.org/en-US/docs/Web/MathML/Element/mmultiscripts;; https://www.tutorialspoint.com/mathml/mathml_matrices.htm.