Topics in Computing Science 162

Computing Science 162: Notes on Big-Oh

An example proof that one function is in the big-Oh of another:

Claim: 16 n² + 5n ∈ O(n²-1)

Proof:
16 n² + 5n ≤ 16 n² + 5n², ∀ n ≥ 1
⇒16 n² + 5n ≤ 31n², ∀ n ≥ 1
⇒16 n² + 5n ≤ 62(n²/2), ∀ n ≥ 1
⇒16 n² + 5n ≤ 62(n²/2 - 1 ) + 62, ∀ n ≥ 1
⇒16 n² + 5n ≤ 62(n²/2 - 1 ) + (n²/2 - 1), ∀ n ≥ 16, since 62 ≤ 127 ≤ (n²/2 -1) when n ≥ 16
⇒16 n² + 5n ≤ 63(n²/2 - 1 ) ∀ n ≥ 16
⇒ 16 n² + 5n ∈ O(n²-1) as witnessed by c=63 and n₀=16.