Please decide whether the following asymptotic notation-related statements are True or False and explain why:
1) 5n^2 = O(n^3)
2) n^2 + 10n = O(n^3)
3) 3n + 1 = O(n)
4) 5n^2 = Ω(n^2)
5) n = Θ(5n^2)
6) 6n^2 + n = Ω(n^3)
7) a^n = O(b^n), b > a > 1
8) lg n = O(n)
9) 10n^2 + 1 ∈ O(n^3)
10) 5n^2 + 10 ∈ Ω(n)
Hint: f(n) = O(g(n))
Asymptotically, f(n) <= g(n)
f(n) = Ω(g(n))
Asymptotically, f(n) >= g(n)
f(n) = Θ(g(n))
Asymptotically, f(n) = g(n)
Please obtain the correct Θ notation for the following expressions:
a) 2(lg n)^2 + 4n + 3n^2lg n
b) (6n^3lg n + 4)(10 + n)
c) ((n^2 + lg n)(n + 1))/(n + n^2)
d) 8^lg n
Hint: drop low order terms and leading coefficient
Using the formal definition of the asymptotic notations, prove the following statements mathematically:
a) 10n^2 + 1 ∈ O(n^3)
b) 5n^2 + 10 ∈ Ω(n)
Hint: find c and n_0