Alice has asked you to help her classify these functions based on their asymptotic growth rate. That is, she wants you to prove a tight big-Theta bound for each function by following these steps. i. Use algebra to simply the function for ease of analysis. Remember the goal of simplification is to isolate terms and identify the dominant term. For instance, we might want to expand (n+1)² to n² +2n+1 to see that n² is the dominant term. ii. Use your intuition, the limit test, or algebra to come up with a good guess of a tight bound. For instance, after isolating n² as the dominant term we might guess our polynomial is in (n²). iii. Use the limit test or the definitions to prove that your guess is correct. Both methods can provide correct proofs, and its a good idea to practice both methods. Sometimes our guess is incorrect and we need to return to the previous step.

Alice has asked you to help her classify these functions based on their asymptotic growth rate. That is, she wants you to prove a tight big-Theta bound for each function by following these steps. i. Use algebra to simply the function for ease of analysis. Remember the goal of simplification is to isolate terms and identify the dominant term. For instance, we might want to expand (n + 1)² to n² +2n+1 to see that n² is the dominant term. ii. Use your intuition, the limit test, or algebra to come up with a good guess of a tight bound. For instance, after isolating n² as the dominant term we might guess our polynomial is in (n²). iii. Use the limit test or the definitions to prove that your guess is correct. Both methods can provide correct proofs, and its a good idea to practice both methods. Sometimes our guess is incorrect and we need to return to the previous step. Transcribed Image Text: Alice has asked you to help her classify these functions based on their asymptotic growth rate.…

Answer the question using Python programming language and Numpy:

N.png ... If gcd(a, m) = 1, then the unique solution xo to ax = b (mod m) is equal to (1) xo = ab (mod m) (2) xo = ab (mod m), where b is the inverse of b. (3) xo = ām (mod m), where ā is the inverse of a. (4) xo = āb (mod m), where ā is the inverse of a. A Choice (2) (B) Choice (1) Choice (4) (D) Choice (3)

Can you represent the following Boolean function f(A, B) with a linear classifier? If yes, show the linear classifier. If not, briefly explain why not. f(A, B) A B 1 1 1 1

For the following pairs of functions, first decide whether f(n) dominates g(n), or g(n) dominates f(n); then decide whether f = O(g), or f = Ω(g), or f = Θ(g), and briefly explains. i. f(n)=n2,g(n)=1000n+30 √1 ii. f(n) = n, g(n) = n3 iii. f(n) = 10 log2 n, g(n) = log10(n3) iv. f(n)=n 100 n ,g(n)=1.2 . 4 nn v. f(n)=2 ,g(n)=2.01 .

Let f (f(n) and g(n)) be asymptotically nonnegative functions. Using the basic definition of Θ notation, prove that max(f(n), g(n)) = Θ(f(n) + g(n)),

A Boolean function F (A,B,C) of truth table at below（PLS step by step） 1，Provide the minterm expansion of F .  2，Provide the minterm expansion and the maxterm expansion of F’s complement, respectively. Assume that we don’t care the value of F when ABC=001. i. What is the minterm expansion of F?  ii. Choose a proper value for the don’t care term and provide the shortest possible expression of POS (product of sum) for F . (Hint: Minimize the number of maxterms)