Prove or disprove the following statements, using the relationship among typical growth-rate functions seen in class....
Prove or disprove the following statements, using the relationship among typical growth-rate functions seen in class.
a)n^15log n + n^9 is O(n^9 log n)
b) 15^7n^5 + 5n^4 + 8000000n^2 + n is Θ(n^3)
c) n^n is Ω (n!)
d) 0.01n^9 + 800000n^7 is O(n^9)
e) n^14 + 0.0000001n^5 is Ω(n^13)
f) n! is O(3n)
Homework Answers
a) There does not exist any positive constant c and n0 such that
0 <= n^15log n + n^9 <= c*n^9 log n
Hence, this statement is false.
b) There does not exist any constant c which always guarantees that
0 <= 15^7n^5 + 5n^4 + 8000000n^2 + n <= c * n3
Hence, this statement is false.
c) Clearly,
1 * 2 * .... * n < n * n * ..... * n
Hence, n! < n^n
This means that n^n is Ω (n!)
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