Assume it is given that T1(n) = O(g1(n)) and T2(n) = O(g2(n)). Prove or disprove each...

Question

Question

Assume it is given that T1(n) = O(g1(n)) and T2(n) = O(g2(n)). Prove or disprove each one of the following claims

T1(n)/T2(n) = O(g1(n)/g2(n))

engineering Computer-Science

Answer 1

Answer #1

f(n) = O(g(n)) if there exists a positive integer n₀ and a positive constant c, such that f(n) <= c.g(n) ∀ n >= n₀

Since T1(n) = O(g1(n)) and T2(n) = O(g2(n)), this means

0 <= T1(n) <= c1 * g1(n) -----------> (1)

0 <= T2(n) <= c2 * g2(n) ------------> (2)

a)

Dividing 1 by 2, we have

0 <= T1(n) / T2(n) <= (c1 / c2) * (g1(n) / g2(n))

0 <= T1(n) / T2(n) <= c3 * (g1(n) / g2(n)) where c3 = c1 / c2

Hence, T1(n) / T2(n) = O(g1(n) / g2(n))

Answer 2

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