Final answer:
The validity of several statements regarding algorithmic complexity. Statements 0(2ⁿ) and 0(2⁽ⁿ⁺¹⁾) grow at the same complexity, If f(n) = 0(g(n)) then g(n) = Omega (f(n)), and 0(log(base 10) n) grows at the same complexity as 0(log (base 2) n) are true, while statement a & e is false. (option b,c & d)
Step-by-step explanation:
The concept of algorithmic complexity is often analyzed using Big O, Big Omega, and other related notations within the field of computer science. Here are the answers to the individual statements:
Statement a: Both O(n!) and O((n-1)!) grow at the same complexity. This is false because n! grows faster than (n-1)!.
Statement b: O(2^n) and O(2^(n+1)) grow at the same complexity. This is true; the complexity is the same because a constant factor does not change the exponential growth rate.
Statement c: If f(n) = O(g(n)) then g(n) = Omega(f(n)). This is true; it's the formal way of saying that g(n) grows at least as quickly as f(n).
Statement d: O(log(base 10) n) grows at the same complexity as O(log(base 2) n). This is true, given that logarithms of different bases are related by a constant factor, and thus do not affect the growth classification in Big O notation.
Statement e: Computer speed does not have any impact on algorithmic complexity. This is false; algorithmic complexity is a theoretical measure that does not depend on the actual speed of the computer, but rather on the growth rate of the operation count as input size increases.