We can extend our notation to the case of two parameters \(n\) and \(m\) that can go to infinity independently at different rates. For a given function \(g(n, m)\), we denote by \(O(g(n, m))\) the set of functions
\[\begin {aligned} O(g(n, m)) = \{ f(n , m) \text { : } & \text {there exist positive constants } c, n_0, \text { and } m_0 \\ & \text { such that } 0 \le f (n, m) \le cg(n, m) \\ & \text { for all } n \ge n_0 \text { or } m \ge m_0\} \end {aligned}\]Give corresponding definitions for \(\Omega(g(n, m))\) and \(\Theta(g(n, m))\).
\(\Omega(g(n, m))\) and \(\Theta(g(n, m))\) can be defined as follows:
\[\begin {aligned} \Omega(g(n, m)) = \{ f(n , m) \text { : } & \text {there exist positive constants } c, n_0, \text { and } m_0 \\ & \text { such that } 0 \le cg(n, m) \le f (n, m) \\ & \text { for all } n \ge n_0 \text { or } m \ge m_0\} \end {aligned}\] \[\begin {aligned} \Theta(g(n, m)) = \{ f(n , m) \text { : } & \text {there exist positive constants } c_1, c_2, n_0, \text { and } m_0 \\ & \text { such that } 0 \le c_1g(n, m) \le f (n, m) \le c_2g(n, m) \\ & \text { for all } n \ge n_0 \text { or } m \ge m_0\} \end {aligned}\]