1. Please decide whether the following asymptotic notation-related statements are True or False, explains why.
1) 5n²=O(n³)
2) n²+10n=O(n³)
3) 3n+1=O(n)
4) 5n²=?(n²)
5) n=O(5n²)
6) 6n²+n=?(n³)
7) a?=O(b?), b>a>1
8) lg n=O(n)
9) 10n²+1 ? O(n³)
10) 5n²+10 ? ?(n)
11) 30n² + 100 ? ?(n³)
Hint:
f(n) = O(g(n))
Asymptotically, f(n) "?" g(n)
f(n) = ?(g(n))
Asymptotically, f(n) "?" g(n)
f(n) = ?(g(n))
Asymptotically, f(n) "=" g(n)
2. Please obtain the correct O notation for the following expressions:
a) 2(lgn)² + 4n + 3n²lgn
b) (6n³lgn + 4) (10+n)
c) [(n²+lgn)(n+1)]/(n+ n²)
d) 8lgn
Hint: drop low order terms and leading coefficient
3. Using the formal definition of the asymptotic notations, prove the following statements mathematically:
a) 10n²+1 ? O(n³)
b) 5n²+10 ? ?(n)
Hint: find c and n?
