In each case determine whether the statement is true or fals

In each case determine whether the statement is true or false. Be sure to explain.

(a) ln(n) = o(n).

(b) 10⁵n³ = o(n⁴).

(c) n² = O(2ⁿ).

Solution

a) ln(n) = O(n)

ln(n) <= kn, the given statement will hold true for k=1, the statement is TRUE since the growth of n is faster than ln(n)

b) 105n3 = O(n4)

10^5n3 <= kn4, the given statement is true for k=105, the statement is TRUE since the growth of n4 is faster than n3

c) n2 = O(2^n)

n2 <= k(2^n), the given statement is TRUE, since there is constant k >1for which the given equation holds true, hence the statement is TRUE