\begin{definition}[Definition:Big-Theta]
'''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:
:$\map f n \in \map \Theta {\map g n}$ 
{{iff}}:
:$\paren {\map f n \in \map \OO {\map g n} } \land \paren {\map f n \in \map \Omega {\map g n} }$
where $\map \OO {\map g n}$ is big-O and $\map \Omega {\map g n}$ is big-omega.
This is read as:
:$\map f n$ is '''big-theta''' of $\map g n$.
Another method of determining the condition is the following limit:
:$\ds \lim_{n \mathop \to \infty} \frac {\map f n} {\map g n} = c$, where $0 < c < \infty$
If such a $c$ does exist, then $\map f n \in \map \Theta {\map g n}$.
Category:Definitions/Order Notation
\end{definition}