Prove or Disprove: For all real valued functions f, if f(n) is O(2¹), then f(2m) is O(2m)

For f (a, b) = (a | b) | b(a) Simplify f (a, b).(b) Find DNF for f (a, b).(c) Is f (a, b) satisfiable?

(1) Find a tight bound solution for the following recurrence: T(m) = T(4) + T(쓱) + cn2 (cis a positive constant) That is, find a function g(n) such that T(n) e O(g(n)). For convenience, you may assume that n is a power of 2, i.e., n=2k for some positive integer k. Justify your answer. [Note: Read question 4-(2) first before writing your answer]

Prove that Note: H(XY) ≤ H(X) using log sum inequality. -X and Y are random variables -X and Y are independent

Problem 4. Let f(n) and g(n) be asymptotically positive functions. Prove or disprove each of the following conjectures. 1. f(n) = O(g(n)) implies g(n) = O(f(n)) 2. f(n)+g(n) = (min(f(n), g(n))) 3. f(n) = (f()) 4. f(n) = O(g(n)) implies g(n) = N(f(n)) 5. f(n) = O((f(n))²)

Find the simplest POS expression using K- Maps for the following functions. For each, list all prime implicants found in the K-Map, then list only the essential prime implicants, and then list the simplest solution. i. f (x1, x2, x3) = II M (0, 1, 2, 4, 5) ii. f (x1,x2, x3, x4) = ПМ (0, 1, 2,4, 5) Em (1, 2,5, 6, 7,8, 10, 14)

Use K-map to obtain the minimized product of sums form of the function f(a,b,c,d) = SEGMA (Q. 1. 2. 3. 4. 6. 8. 9. 10. 11. 3, 15). Use the editor to format your answer

Let x and y be integers such that x = 3 (mod 10) and y = 5 (mod 10). Find the integer z such that 97x + 3y³ z (mod 10) and 0 ≤ z ≤9.

Please answer question.

1. Consider the following functions of n. (1) (2) (3) (4) (5) (6) (7) (8) (9) fi(n) = n, if n = 2.k for k = 0, 1, 2,... = n², otherwise. f2(n) = n, if n = 3.k for k= 0, 1, 2,... = n², otherwise. f3(n) = log n. Answer the following questions and prove your answ Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. Yes or No. fi(n) = 2(f₂(n)), f2(n) = Q(fi(n)), fi(n) = 0 (f₂(n)), f₂(n) = 0 (fi(n)), fi(n) = O(n), f₂(n) = O(n²), f3(n) = O(√n), f3(n) = O(fi(n)), 22n = 0(2¹), swer for each one.