# Symbols & Notations

Notice: This page is a work in progress. It is constantly being updated.

Page last updated: 20.10.2016

Symbol / Notation Meaning
$=$ is equal to …
$\neq$ is not equal to…
$\equiv$ is identical or congruent to …
$\approx$ is approximately equal to …
$\cong$ is isomorphic to …
$\propto$ is proportional to …
$<$ is less than …
$\le$ is less than or equal to …
$\nless$ is not less than …
is greater than …
$\ge$ is greater than or equal to …
$\ngtr$ is not greater than …
$x\wedge y$ x and y
$x\vee y$ x or y (or both)
$x\Rightarrow y$ x implies that y (if x then y)
$x\Leftarrow y$ x is implied by y (if y then x)
$x\Leftrightarrow y$ x implies and is implied by y (x is equivalent to y)
$\sim x$ not x
$\in$ is an element of …
$\notin$ is not an element of …
$\exists$ there exists …
$\forall$ for all …
$\infty$ infinity
$\left\{ { x }_{ 1 },{ x }_{ 2 },{ x }_{ 3 },... \right\}$ a set with elements ${ x }_{ 1 },{ x }_{ 2 },{ x }_{ 3 },...\quad$
$\left\{ x:... \right\}$ the set of all x such that …
$n\left( A \right)$ the number of elements in set A
$\emptyset ,\{ \}$ an empty set
$\xi$ the universal set
$A$ the complement of the set A
$\subseteq$ is a subset of …
$\supseteq$ is a superset of …
$\subset$ is a proper subset of …
$\cup$ union
$\cap$ intersection
$\mathbb{N}$ the set of natural numbers, $\left\{ 1,2,3,4,... \right\}$
$\mathbb{Z}$ the set of integers, $\left\{ 0,\pm 1,\pm 2,\pm 3,... \right\}$
$\mathbb{Z}^+$ the set of positive integers, $\left\{ 1,2,3,4,... \right\}$
$\mathbb{Z}_n$ the set of integers modulo n, $\left\{ 1,2,3,4,..,n-1 \right\}$
$\mathbb{Q}$ the set of rational numbers, $\left\{ \frac { a }{ b } :a\in \mathbb{Z}_u,\quad b\in \mathbb{Z}^+ \right\}$
$\mathbb{Q}^+$ the set of positive rational numbers,
$\mathbb{Q}_0^+$ the set of positive rational numbers including zero, $\{ x\in \mathbb{Q}:x\ge 0\}$
$\mathbb{R}$ the set of real numbers
$\mathbb{R}^+$ the set of positive real numbers,
$\mathbb{R}_0^+$ the set of positive real numbers including zero, $\{ x\in \mathbb{R}:x\ge 0\}$
$\mathbb{C}$ the set of complex numbers
[p,q] a closed interval, $\left\{ x\in \mathbb{R}:p\le x\le q \right\}$
[p,q), [p,q[ the interval, $\left\{ x\in \mathbb{R}:p\le x
(p,q], ]p,q] the interval, $\left\{ x\in \mathbb{R}:p
(p,q), ]p,q[ an open interval, $\left\{ x\in \mathbb{R}:p
${ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+...+{ a }_{ n }$
${ a }_{ 1 }\times { a }_{ 2 }\times { a }_{ 3 }\times ...\times { a }_{ n }$
$\sqrt { n }$ the positive square root of n
$\left| n \right|$ the modulus of n
n! $1\times 2\times 3\times ...\times \left( n-2 \right) \times \left( n-1 \right) \times n$
$\left( \begin{matrix} n \\ k \end{matrix} \right)$
$f\left( x \right)$ the function of x
${ f }^{ -1 }$ the inverse of the function
$\Delta x,\quad \delta x$ change in x
$\frac { dy }{ dx }$ the derivative of y with respect to x
$\frac { { d }^{ n }y }{ d{ x }^{ n } }$ the nth derivative of y with respect to x
$f$ the first, second, …, and nth derivatives of the function of x with respect to x
$\int { ydx }$ y integrated
$\int _{ a }^{ b }{ ydx }$ y integrated between the limits a and b
$\frac { \partial A }{ \partial x }$ the partial derivative of A with respect to x
$e$ base of natural logarithms, with a value of approximately 2.718
${ e }^{ x },exp\quad x$ exponential function of x
$\log _{ a }{ x }$ logarithm of x to the base a
$\ln { x } ,\log _{ e }{ x }$ the natural logarithm of x