Let S(n) = f(1) + f(2) + ... + f(n). For each f(n) below, derive the...
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3} n times). (a) What is Sl? Give a brief explanation. (b) For 0 <k <n, let T be the set of all elements of S with exactly k occurrences of 3's. Determine |Tx I, and prove it using a bijection. In your solution, you need to define a set Ax that involves subsets and/or cartesian products with known cardinalities. Then clearly define your bijection...
For each pair of functions f(n) and g(n), indicate whether f(n) = O(g(n)), f(n) = Ω(g(n)), and/or f(n) = Θ(g(n)), and provide a brief explanation of your reasoning. (Your explanation can be the same for all three; for example, “the two functions differ by only a multiplicative constant” could justify why f(n) = n, g(n) = 2n are related by big-O, big-Omega, and big-Theta.) i. f(n) = n^2 log n, g(n) = 100n^2 ii. f(n) = 100, g(n) = log(log(log...
2 (25 pts). Let an algorithm has complexity S(n)=S(n-1)+f(n), where for k=1,2,3,... f(k)=k+k/3. Answer these two questions: (1) Find the closed form for S(n) if S(2)=1. (2) Prove by mathematical induction that the closed form you found is correct.
1. For each of the following pairs of functions, prove that f(n)-O(g(n)), and / or that g(n) O(f(n)), or explain why one or the other is not true. (a) 2"+1 vs 2 (b) 22n vs 2" VS (c) 4" vs 22n (d) 2" vs 4" (e) loga n vs log, n - where a and b are constants greater than 1. Show that you understand why this restriction on a and b was given. f) log(0(1) n) vs log n....
Let f(n) = n^2 +200 Let g(n) = 200 n Select the first answer below that is true. f is Theta (g) f is O (g) f is Ohm (g)
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) = 0.5n3 . Prove that f(n) = O(g(n)) using the definition of Big-O notation. (You need to find constants c and n0). b) Let f(n) = 3n2 + n and g(n) = 2n2 . Use the definition of big-O notation to prove that f(n) = O(g(n)) (you need to find constants c and n0) and g(n) = O(f(n)) (you need to find constants c and n0). Conclude that...
Below are sample questions: [5] 6. Let X F (V1, V2) where v2 > 2. Derive E(X) = 2. Show your work. Hint: You may use the result that if Y ~ (v), then E(Y") = 2 r>-v/2. ru2 + 2/4 for
For this question, let S be a sample space, and let RV be the set of {0, 1}-valued random variables. Let F : RV → (2^S) be given by F(X) := (X = 1). Let I : (2^S) → RV be the function that outputs the indicator variable for A on input A. Show that I and F are two-sided inverses. Note: 2^S denotes power set of S
This Question: 1 pt 2 of 3 Let f(x) = x + 2 and g(x)=x²-x. Find and simplify the expression. (f+g)(2) (f+9)(2) = (Simplify your answer.)
1. Let (S;F;P) be a probability space with A 2 F and B 2 F such that P(A) = 0:3 and P(B) = 0:4. Find the following probabilities under the specified conditions. Note that I don’t expect you to have to show much work in answering this question. (a) either A or B occurs if A and B are mutually exclusive (b) either A or B occurs if A and B are statistically independent (c) either A or B occurs...