# Definition:Big-Omega Notation

## Definition

### Definition 1

Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.

Then $\map \Omega g$ is defined as:

- $\map \Omega g = \set {f: \N \to \R: \exists c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le c \cdot \size {\map g n} \le \size {\map f n} }$

### Definition 2

Let $f: \N \to \R, g: \N \to \R$ be two real sequences, expressed here as real-valued functions on the set of natural numbers $\N$.

Let there exist $c \in \R_{>0}$ such that:

- $\ds \lim_{n \mathop \to \infty} {\frac {\map f n} {\map g n} } = c > 0$

Then:

- $\map f n \in \map \Omega {\map g n}$

## Notation

The expression $\map f n \in \map \Omega {\map g n}$ is read as:

**$\map f n$ is big-omega of $\map g n$**

While it is correct and accurate to write:

- $\map f n \in \map \Omega {\map g n}$

it is a common abuse of notation to write:

- $\map f n = \map \Omega {\map g n}$

This notation offers some advantages.

## Also known as

Note that in the Greek alphabet, $\Omega$ is the **capital letter** counterpart of the **minuscule** $\omega$.

Hence the former is called **big-omega** and the latter **little-omega**.

Some sources, therefore, write **$\Omega$ notation** as **big-$\Omega$ notation**, despite the fact that $\Omega$'s "big"-ness is intrinsic.

$\mathsf{Pr} \infty \mathsf{fWiki}$ may sometimes adopt this convention if clarity is improved.

## Motivation

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

## Also see

- Results about
**big-$\Omega$ notation**can be found**here**.