Q Sum For parts (a)-(e), decide if it is true or false. If the statement is true, then prove it (to prove it you will either need to use the definition and find constants or use the limit laws). If the statement is false, then give a reason or a counterexample a. 10n2 = O(n3). b. n2 ∈ Ω(n3) c. 2n ∈ O(2n+1) d. n! ∈ Θ((n + 1)!) e.√n = O(nsin(n))

es remaining 8. Consider the function f:NxN-N defined recursively by: 1) Base case: Let meN and define (0,m) = 0 2) Recursive case: For any x,meN, x>0, define f(x,m) = (x-1,m) + (m+m) Prove the following theorem holds using proof by induction: Thereom: For any n,meN, m>0 we have (n.m) I m = n+n Fill in your answer here 9 Help BIU X, x L - ɔE =N E X Format

Q7. The big O bound for the following functions is T(n)= nlogn2+ nlogn OO (log(log n)) OO (nlogn) 00 (n² log n) 00 (n²)

Let f (n) and g(n) be positive functions (for any n they give positive values) and f (n) = O(g(n)).Prove or disprove the following statement:

I'm not sure how to use pumping lemma on this or if I should be using a different method

7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).

Please answer it asap. both questions.

You know that the truth value of the statement (p q) → r is Truth. What can you say about the truth value of the following statement: ~r→ (~ pV ~ q) OTHER False cannot be decided Truth

don't use others answers

Let f (n) and g(n) be functions with domain {1, 2, 3, . . .}. Prove the following: If f(n) = O(g(n)), then g(n) = Ω(f(n)).