Big O notation

[math]\displaystyle{ \Theta(g(n)) = \{f(n) : }[/math] there exist positive constants [math]\displaystyle{ c_1 }[/math],[math]\displaystyle{ c_2 }[/math] and [math]\displaystyle{ n_0 }[/math] such that [math]\displaystyle{ 0\le \;c_1g(n) \le \; f(n) \le \; c_2g(n) }[/math] for all [math]\displaystyle{ n \ge \; n_0 \} }[/math]

[math]\displaystyle{ O(g(n)) = \{f(n) : }[/math] there exist positive constants [math]\displaystyle{ c }[/math] and [math]\displaystyle{ n_0 }[/math] such that [math]\displaystyle{ 0\le \; f(n) \le \; cg(n) }[/math] for all [math]\displaystyle{ n \ge \; n_0 \} }[/math]

[math]\displaystyle{ \Omega(g(n)) = \{f(n) : }[/math] there exist positive constants [math]\displaystyle{ c }[/math] and [math]\displaystyle{ n_0 }[/math] such that [math]\displaystyle{ 0 \le \; cg(n)\le \; f(n) }[/math] for all [math]\displaystyle{ n \ge \; n_0 \} }[/math]

[math]\displaystyle{ o(g(n)) = \{f(n) : }[/math] for any positive constant [math]\displaystyle{ c \gt 0 }[/math], there exists ac constant [math]\displaystyle{ n_0 \gt 0 }[/math] such that [math]\displaystyle{ 0 \le \; f(n) \lt cg(n) }[/math] for all [math]\displaystyle{ n \ge \; n_0 \} }[/math]