asymtotic analysis

Intuition:
O: \leq \theta: = \Omega: \geq Definitions:
f(n) = O(g(n)) \implies \exists n_{0}: \forall n > n_0, f(n) \leq c (g(n)) f(n) = \Omega(g(n)) \implies \exists n_{0}: \forall n > n_0, f(n) \geq c (g(n)) f(n) = \theta(g(n)) \implies \exists n_{0}: \forall n > n_0, f(n) \geq 1 (g(n)), f(n) \leq c (g(n)) ~ Given functions f(n) and g(n), if:

\begin{equation} \lim_{n\to \infty} \left(\frac{f(n)}{g(n)}\right) = 1 \end{equation}

we say that f \sim g.
That – the relationship between f and g grows in a similar fashion as n increases. For instance:
f(n) = n+1 g(n) = n+2 Therefore:

\begin{equation} f\sim g = \lim_{n\to \infty} \frac{f(n)}{g(n)} = \lim_{n\to \infty} \frac{n+1}{n+2} = 1 \end{equation}

The \sim operator is commutative (f \sim g \Rightarrow g\sim f) and transitive (f\sim g, g\sim h \Rightarrow f \sim h).
o(n) Given two functions f(n), g(n), if their relationship shows:

\begin{equation} \lim_{n \to \infty} \frac{f(n)}{g(n)} = 0 \end{equation}

we can write it as

\begin{equation} f = o(g) \end{equation}

This tells us that if n becomes very large, g becomes much larger than f. f does not grow nearly as fast as g.
The operation is not commutative, but is transitive (f = o(g), g = o(h) \Rightarrow f = o(h))
O(n) Given two functions f(n), g(n).

\begin{equation} \lim_{n \to \infty} \frac{f(n)}{g(n)} < \infty \end{equation}

that the relationship between f(n) and g(n) is countable as n trends to infinity.
We can also say that, given n, n_0, and some c which \forall n, n > n_0, there is:

\begin{equation} |f(n)| < |cg(n)| \end{equation}

This tells us that f(n) does not grow much much faster than g(n).
Therefore:
If f \sim g, f = O(g) (as they grow together, f is not much faster) If f = o(g), f=O(g) (as f does not grow at all, f is not faster) \theta(n) f=\theta(g) IFF f=O(g) and g=O(f), its essentially \sim but without the strict requirement of a 1:1 ratio.
\omega(n) and \Omega(n) The inverses of O and o:
f(n) = O(g(n)) \Rightarrow g(n) = \omega(f(n)) f(n) = o(g(n)) \Rightarrow g(n) = \Omega(f(n))