# Symbols & Notations

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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