# Definition:Order Notation

## Definition

### $O$-Notation

$\OO$ notation is a type of order notation, typically used in computer science for comparing 'run-times' of algorithms, or in analysis for comparing growth rates between two growth functions.

### $\Theta$-Notation

Big-theta notation is a type of order notation for typically comparing 'run-times' or growth rates between two growth functions.

Big-theta is a stronger statement than big-O and big-omega.

Suppose $f: \N \to \R, g: \N \to \R$ are two functions.

Then:

$\map f n \in \map \Theta {\map g n}$
$\paren {\map f n \in \map \OO {\map g n} } \land \paren {\map f n \in \map \Omega {\map g n} }$

where $\map \OO {\map g n}$ is big-O and $\map \Omega {\map g n}$ is big-omega.

$\map f n$ is big-theta of $\map g n$.

### $\Omega$-Notation

Big-Omega notation is a type of order notation for typically comparing 'run-times' or growth rates between two growth functions.

Let $f, g$ be two functions.

Then:

$\map f n \in \map \Omega {\map g n}$
$\exists c > 0, k \ge 0: \forall n > k: \map f n \ge c \map g n$

$\map f n$ is big omega of $\map g n$.

### $\omega$-Notation

Let $f$ and $g$ be real functions.

Then:

$\map f n \in \map \omega {\map g n}$

is equivalent to:

$\ds \lim_{n \mathop \to \infty} {\frac {\map f n} {\map g n} } = \infty$