Definition:Theta Notation/Definition 1

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

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


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

where:
 * $\map \OO g$ is big-$\OO$ notation
 * $\map \Omega g$ is big-$\Omega$ notation.

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

Also presented as
This definition can also be presented as:


 * $\map f n \in \map \Theta {\map g n}$


 * $\paren {\map f n \in \map \OO {\map g n} } \text { and } \paren {\map f n \in \map \Omega {\map g n} }$
 * $\paren {\map f n \in \map \OO {\map g n} } \text { and } \paren {\map f n \in \map \Omega {\map g n} }$

Also see

 * Equivalence of Definitions of Theta Notation