かじもとにっき

ゆるふわなおはなしをかきます

MathJaxのテスト

\[\begin{array}{l}O(g) = \left\{ {f:\exists c > 0.\exists {n_0}.\forall n \ge {n_0}.f(n) \le cg(n)} \right\}\\\Omega (g) = \left\{ {f:\exists c > 0.\exists {n_0}.\forall n \ge {n_0}.f(n) \ge cg(n)} \right\}\\\Theta (g) = O(g) \cap \Omega (g)\end{array}\]

\[\begin{array}{l}f \in \Theta (g)\\ \Leftrightarrow \left( {\exists c > 0.\exists {n_0}.\forall n \ge {n_0}.f(n) \le cg(n)} \right) \wedge \left( {\exists c > 0.\exists {n_0}.\forall n \ge {n_0}.f(n) \ge cg(n)} \right)\\ \Leftrightarrow \exists c > 0.\exists c' > 0.\exists {n_0}.\exists {{n'}_0}.\left( {\forall n \ge {{n'}_0}.f(n) \le c'g(n)} \right) \wedge \left( {\forall n \ge {n_0}.f(n) \ge cg(n)} \right)\\ \Leftarrow \exists c > 0.\exists c' > 0.\exists {n_0}.\left( {\forall n \ge {n_0}.f(n) \le c'g(n)} \right) \wedge \left( {\forall n \ge {n_0}.f(n) \ge cg(n)} \right)\\ \Leftrightarrow \exists c > 0.\exists c' > 0.\exists {n_0}.\forall n \ge {n_0}.cg(n) \le f(n) \le c'g(n)\end{array}\]