The function f is defined for non-negative integers a and b recursively as follows: f(a, b) ={ 0     if                                   a = 0 or                      b = 0 f(a − 1, b − 1) + 2a − 1       if a = b f(a − b, b) + f(b, b)              if a > b f(a, a) + f(b − a, a)               if a < b } Compute f (3, 2) by drawing a recursion tree showing all of the computation required and then use your tree to compute the answer.

8- Determine if each of the following recursive definition is a valid recursive definition of a function f from a set of non-negative integers. If f is well defined, find a formula for f(n) where n is non- negative and prove that your formula is valid. a. f(0) = 2,f(1) = 3, f(n) = f(n-1)-1 for n ≥ 2 b. f(0) = 1,f(1) = 2, f(n) = 2f (n-2) for n = 2

= = 2×2 and 6 = (a) A composite number is a positive integer that has at least one divisor other than 1 and itself. For example, 2 1×2 is not a composite number but 4 2 × 3 are composite numbers. A logic circuit has four binary input variables, A, B, C and D. The output Z of the logic circuit is 1 if the unsigned integer represented by the binary number ABCD is a composite number. Using variables A and B for the select inputs S1 and S0 of a 4-to-1 multiplexer, implement the logic function Z(A, B, C, D) using this multiplexor and other logic gates.

a. Correctness of dynamic programming algorithm: Usually, a dynamic programming algorithm can be seen as a recursion and proof by induction is one of the easiest way to show its correctness. The structure of a proof by strong induction for one variable, say n, contains three parts. First, we define the Proposition P(n) that we want to prove for the variable n. Next, we show that the proposition holds for Base case(s), such as n = 0, 1, . . . etc. Finally, in the Inductive step, we assume that P(n) holds for any value of n strictly smaller than n' , then we prove that P(n') also holds. Use the proof by strong induction properly to show that the algorithm of the Knapsack problem above is correct.    b. Bounded Knapsack Problem: Let us consider a similar problem, in which each item i has ci > 0 copies (ci is an integer). Thus, xi is no longer a binary value, but a non-negative integer at most equal to ci , 0 ≤ xi ≤ ci . Modify the dynamic programming algorithm seen at class for this…

Suppose a recursive function f(n)=3f(n-1) + 2. If f(0)=1, what is f(2)? 53 17 13 O O O O

Can you please help me solve problem 2?

In any CAS or programmng language

Let recursive algorithm F be defined as follows: F(int n):if n=1 then    return 1else   return F(n-1)*n (a) i. Assuming x is a positive integer, what mathematical expression is returned by F(x)?ii. Repeat (i) but assume the line “return 1” is replaced by “return 0”? (b) Which of the following replacements for the last line will yield a function guaranteed to terminate for all valid inputs? i. return F(n-2)+2ii. return F( )*2iii. return F( )*2

Using recursion, write a Python function def before(k,A) which takes an integer k and an array A of integers as inputs and returns a new array consisting of all the integers in A which come before the last  occurrence of k in A, in the same order they are in A. For example, if A is [1,2,3,6,7,2,3,4] then before(3,A) will return [1,2,3,6,7,2]. If k does not occur in A, the function should return None.

Program #2 Define a sequence of numbers recursively Define a sequence ao, a1, az, a3, where ao = 1 an = (an -1+ 1) * 2 an = (2* an - 2+ an-1) ifn is even So for instance: (n = 0) ao = 1 (n = 1) ai = (ao + 1) *2 = (1+ 1) *2= 4 (n = 2) az = (2* ao + ao) = 2*1+4 = 6 (n = 3) az = (az + 1) *2 = (6 + 1) *2 =14 The resulting sequence is 1, 4, 6, 14, . if n is odd !3! %3D You will write a function that returs the nth term of the sequence. You must do this by using a recursive function. The function will NOT print out any values. Rather, the function will return the nth term of the sequence using a recursive algorithm. In the main part of the program: 1. Input the number of terms of the sequence to output. 2. In a for loop, call the function repeatedly to get the desired number of terms. The function will take i (assuming the for index is called i) as the argument and return the ith term of the sequence. 3. In the for loop, print out each term as it is returned. Figure out the necessary prompts…