Definition:Theta Notation/Definition 1

Definition
Let $g: \N \to \R$ be a real function.

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


 * $\map \Theta g = \map \OO g \cap \map \Omega g$

where $\map \OO g$ is big-$\OO$ and $\map \Omega g$ is big-$\Omega$.

That is:
 * $\map \Theta g = \set {f: \exists c_1, c_2 \in \R_{>0}: \exists n_0 \in \N: \forall n > n_o: 0 \le c_1 \cdot \size {\map g n} \le \size {\map f n} \le c_2 \cdot \size {\map g n} }$

This is read as:
 * $\map f n$ is theta of $\map g n$.