Mathematical notation and terminology¶

Sets¶

Symbol	Reads	Explanation	Example
$\{ \ \dots \}$	a set	The elements of the set are written inside the braces.	$\{1,2,3\}$ denotes the set consisting of the numbers 1, 2 and 3.
${ \dots	\dots }$	The set of all $\dots$ satisfying the condition $\dots$.	This denotes the set consisting of all objects satisfiying a certain condition.
$\in$	is an element of	If $M$ is a set the expression $x \in M$ means that $x$ is a member of $M$.	$\diamondsuit \in \{ \diamondsuit, \heartsuit, \spadesuit, \clubsuit \}$
$\notin$	is not an element of	If $M$ is a set the expression $x \in M$ means that $x$ is a member of $M$.	$\diamondsuit \notin \{ \heartsuit, \spadesuit, \clubsuit \}$

$f : X \to Y$	$f$ from $X$ to $Y$	A function $f$ from a set $X$ to another set $Y$.	$f : \{ \text{Monday}, \dots, \text{Sunday} \} \to \{ \text{true, false} \}$ is some function that assigns to any weekday either true or false. For example, $f$ could indicate whether I go to school that day.
$\to$	to	The regular arrow is the symbol for a function.
$\mapsto$	maps to	$x \mapsto y$ indicates that a particular element $x \in X$ is sent to (or “mapped to”) the element $y \in Y$.	$\text{Sunday} \mapsto \text{false}$
$X \times Y$	The product of two sets $X$ and $Y$.	The product consists of pairs $(x, y)$, where $x \in X$ and $y \in Y$.	$\{0, 1\} \times \{ 0, 1\} = \{(0,0), (0, 1), (1,0), (1,1)\}$.
$X \subset Y$	subset	$X$ is a subset of $Y$ if every element of $X$ is also an element of $Y$.	$\{1, 2\} \subset \{0, 1, 2\}$
$X \subsetneq Y$	proper subset	$X$ is a proper subset of $Y$ if $X \subset Y$ but $X \ne Y$	$\{1, 2\} \subsetneq \{0, 1, 2\}$
$X \cap Y$	intersection of $X$ and $Y$	The intersection consists of those elements that are contained in $X$ and in $Y$. $\{0, 1\} \cap \{-1, 0\} = \{ 0 \}$
$X \cup Y$	union of $X$ and $Y$	The union consists of those elements that are contained in $X$ or in $Y$. $\{0, 1\} \cap \{-1, 0\} = \{-1, 0, 1 \}$
$g \circ f$	composition	If $f : X \to Y$ and $g : Y \to Z$ are two functions, then $g \circ f: X \to Z$ is the function sending $x \in X$ to $g(f(x))$.

Logic¶

Symbol	Reads	Explanation	Example
$\Rightarrow$	Implies	If $A$ and $B$ are two (mathematical) statements, then “$A \Rightarrow B$” means that if $A$ holds then $B$ also holds.	$x \ge 1 \Rightarrow x^2 \ge 1$
$\Leftrightarrow$	Equivalent	If $A$ and $B$ are two mathematical statements, then “$A \Leftrightarrow B$” is an abbreviation for $A \Rightarrow B$ and (at the same time) $B \Rightarrow A$.	$x \ge 0$ $\Leftrightarrow$ $x + 1 \ge 1$
$:=$	is defined to be		$x := 2$ means that we define the variable $x$ to take the value 2

Numbers and arithmetic¶

Symbol	Reads	Explanation	Example
${{\bf Z}}$		The set of all integers.	$-34, -1, 0, 1, 2, 18, \dots \in {{\bf Z}}$, $\frac 3 4 \notin {{\bf Z}}$
${{\bf Q}}$		The set of all rational numbers.	$\frac {-3}{16}, -3.3, -1, 0, 2.4, \frac 3 4 \in {{\bf Q}}$, $\sqrt 3 \notin {{\bf Q}}$
${\bf R}$		The set of all real numbers.	0, 1, $-1$, $\frac 1 2$, $\sqrt 3$, $\pi$, $e \in {\bf R}$
$\sum_{e=1}^n a_e$	Sum	This is an abbreviation for the sum of the $a_e$, where $e$ runs from 1 to $n$. (Here $a_e$ can be any expression depending on $e$.) It can also be written as $a_1 + a_2 + \dots + a_e$.	$\sum_{e=1}^3 e^2 = 1^2 + 2^2 + 3^2 = 14$.

Trigonometric functions¶

Angles can be measured in degrees or in radians. These are converted as follows:

angle	radian
(in degree)	(no unit)
$180^\circ$	$\pi$
$90^\circ$	$\frac \pi 2$
$\alpha$	$\frac{\pi}{180} \alpha$
$\frac {180}\pi r$	$r$

Geometrically, given an angle $\alpha$ (between 0 and $360^\circ$ as in the picture below), the radian is the length of the yellow circle segment as shown:

A rotation by a positive number is counter-clockwise; conversely negative numbers correspond to a clockwise rotation. For example, a rotation by $\frac \pi 2$ is a counter-clockwise rotation by $90^\circ$. A rotation by $-\frac \pi 4$ is a clockwise rotation by $45^\circ$.

Given any radian $r$, the ray that has an angle $r$ between itself and the positive $x$-axis meets the circle with radius 1 and mid-point $(0,0)$ in exactly one point $p$. The trigonometric functions $\sin$ and $\cos$ are defined to be the coordinates of that point:

\[ p = (\cos(r), \sin (r)). \]

For example, we have the following values

$r$	0	$\pi/6$ (30°)	$\pi/4$ (45°)	$\pi/3$ (60°)	$\pi/2$ (90°)	$\dots$
$\sin(r)$	0	$\frac 12$	$\sqrt 2$	$\frac {\sqrt 3}2$	1	$\dots$
$\cos(r)$	1	$\frac {\sqrt 3} 2$	$\sqrt 2$	$\frac 12$	0	$\dots$

Symbol	Reads	Explanation	Example
\(\{ \ \dots \}\)	a set	The elements of the set are written inside the braces.	\(\{1,2,3\}\) denotes the set consisting of the numbers 1, 2 and 3.
${ \dots	\dots }$	The set of all \(\dots\) satisfying the condition \(\dots\).	This denotes the set consisting of all objects satisfiying a certain condition.
\(\in\)	is an element of	If \(M\) is a set the expression \(x \in M\) means that \(x\) is a member of \(M\).	\(\diamondsuit \in \{ \diamondsuit, \heartsuit, \spadesuit, \clubsuit \}\)
\(\notin\)	is not an element of	If \(M\) is a set the expression \(x \in M\) means that \(x\) is a member of \(M\).	\(\diamondsuit \notin \{ \heartsuit, \spadesuit, \clubsuit \}\)

\(f : X \to Y\)	\(f\) from \(X\) to \(Y\)	A function \(f\) from a set \(X\) to another set \(Y\).	\(f : \{ \text{Monday}, \dots, \text{Sunday} \} \to \{ \text{true, false} \}\) is some function that assigns to any weekday either true or false. For example, \(f\) could indicate whether I go to school that day.
\(\to\)	to	The regular arrow is the symbol for a function.
\(\mapsto\)	maps to	\(x \mapsto y\) indicates that a particular element \(x \in X\) is sent to (or “mapped to”) the element \(y \in Y\).	\(\text{Sunday} \mapsto \text{false}\)
\(X \times Y\)	The product of two sets \(X\) and \(Y\).	The product consists of pairs \((x, y)\), where \(x \in X\) and \(y \in Y\).	\(\{0, 1\} \times \{ 0, 1\} = \{(0,0), (0, 1), (1,0), (1,1)\}\).
\(X \subset Y\)	subset	\(X\) is a subset of \(Y\) if every element of \(X\) is also an element of \(Y\).	\(\{1, 2\} \subset \{0, 1, 2\}\)
\(X \subsetneq Y\)	proper subset	\(X\) is a proper subset of \(Y\) if \(X \subset Y\) but \(X \ne Y\)	\(\{1, 2\} \subsetneq \{0, 1, 2\}\)
\(X \cap Y\)	intersection of \(X\) and \(Y\)	The intersection consists of those elements that are contained in \(X\) and in \(Y\). \(\{0, 1\} \cap \{-1, 0\} = \{ 0 \}\)
\(X \cup Y\)	union of \(X\) and \(Y\)	The union consists of those elements that are contained in \(X\) or in \(Y\). \(\{0, 1\} \cap \{-1, 0\} = \{-1, 0, 1 \}\)
\(g \circ f\)	composition	If \(f : X \to Y\) and \(g : Y \to Z\) are two functions, then \(g \circ f: X \to Z\) is the function sending \(x \in X\) to \(g(f(x))\).

Symbol	Reads	Explanation	Example
\(\Rightarrow\)	Implies	If \(A\) and \(B\) are two (mathematical) statements, then “\(A \Rightarrow B\)” means that if \(A\) holds then \(B\) also holds.	\(x \ge 1 \Rightarrow x^2 \ge 1\)
\(\Leftrightarrow\)	Equivalent	If \(A\) and \(B\) are two mathematical statements, then “\(A \Leftrightarrow B\)” is an abbreviation for \(A \Rightarrow B\) and (at the same time) \(B \Rightarrow A\).	\(x \ge 0\) \(\Leftrightarrow\) \(x + 1 \ge 1\)
\(:=\)	is defined to be		\(x := 2\) means that we define the variable \(x\) to take the value 2

Symbol	Reads	Explanation	Example
\({{\bf Z}}\)		The set of all integers.	\(-34, -1, 0, 1, 2, 18, \dots \in {{\bf Z}}\), \(\frac 3 4 \notin {{\bf Z}}\)
\({{\bf Q}}\)		The set of all rational numbers.	\(\frac {-3}{16}, -3.3, -1, 0, 2.4, \frac 3 4 \in {{\bf Q}}\), \(\sqrt 3 \notin {{\bf Q}}\)
\({\bf R}\)		The set of all real numbers.	0, 1, \(-1\), \(\frac 1 2\), \(\sqrt 3\), \(\pi\), \(e \in {\bf R}\)
\(\sum_{e=1}^n a_e\)	Sum	This is an abbreviation for the sum of the \(a_e\), where \(e\) runs from 1 to \(n\). (Here \(a_e\) can be any expression depending on \(e\).) It can also be written as \(a_1 + a_2 + \dots + a_e\).	\(\sum_{e=1}^3 e^2 = 1^2 + 2^2 + 3^2 = 14\).

angle	radian
(in degree)	(no unit)
\(180^\circ\)	\(\pi\)
\(90^\circ\)	\(\frac \pi 2\)
\(\alpha\)	\(\frac{\pi}{180} \alpha\)
\(\frac {180}\pi r\)	\(r\)

\(r\)	0	\(\pi/6\) (30°)	\(\pi/4\) (45°)	\(\pi/3\) (60°)	\(\pi/2\) (90°)	\(\dots\)
\(\sin(r)\)	0	\(\frac 12\)	\(\sqrt 2\)	\(\frac {\sqrt 3}2\)	1	\(\dots\)
\(\cos(r)\)	1	\(\frac {\sqrt 3} 2\)	\(\sqrt 2\)	\(\frac 12\)	0	\(\dots\)