Question

(Embedded System)

Prove the following statements:

• 10n³∈ Ω(n²)
• 10n²∈ ⊖(n²)

Answer 1

1. Prove

Definition-

Ω (g(n)) = {f(n): there exist positive constants c and n0 such that 0 <= c*g(n) <= f(n) for all n >= n0}. =>

If n=1, c = 1, then

=> true

And

Which is always true =>

2. Prove

Definition-

Θ(g(n)) = {f(n): there exist positive constants c1, c2 and n0 such that 0 <= c1*g(n) <= f(n) <= c2*g(n) for all n >= n0}

1. consider 0 <= c1*g(n) <= f(n),

If c1 = 10, then

is true for all n >= 0

2. Consider 0 <= f(n) <= c2*g(n)

If c2 = 10, then

is true for all n >= 0

Hence both the conditions are true for c1 and c2 >= 10.

=>