1. For the function defined recursively by f(0)=5 and f(n)= 4f (n-1)+3, answer the following: a. Find a closed form representation for this function. Your closed form should not include any series.
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- Determine whether the proposed definition isa valid recursive definition of a function f from the setof nonnegative integers to the set of integers. If f is welldefined, find a formula for f(n) when n is a nonnegativeinteger and prove that your formula is valid. f(0) = 1, f(n) = −f(n − 1) for n ≥ 1Find a closed form representation for the following recursively defined function. Give the run-time complexity of each recursively defined function.7. Find a closed form representation for the function defined recursively by f(1)-10 and f(n)=5ƒ(%) + n.
- Problem 1. Prove that the following functions are Primitive Recursive. I – 1 if x > 0, (1) mPred(x) = for x € N. if x = 0. if x > 0, (2) sgn(x) = for x E N. 0 if x = 0. if x > 0, (3) sgn(x) = for x E N. 1 if x = 0.Implement the XOR function exclusively through a combination of AND, OR, and NOTperceptrons introduced in class. (Note: Non-linear activation functions are prohibited at thispoint.) Also, provide a detailed explanation of your implementation with a summary of your implementation, theresults, and your analysis.7.) Given the following Truth Table for the function F, generate a Karnaugh-Map for the function F (1 Point), and a Boolean Algebra representation of the function F (1 Point). Each variable (A,B,C,D) and its complement (Ā,B, C,D) are available as an input if needed. I provided a blank K-Map template to save you time. A BC D F 1 0 0 01 0 01 0 0 0 1 1 0 10 0 0 1 0 1 0 11 0 01 1 |1 10 0 0 10 0 1 1 0 1 1 10 10 1 |1 110 0 1 10 1 11 1 0 1 1 1 1 1 1
- Construct a subprogram that solves the differential equation set dy g(y,t)dt dt with the fourth-order Runge-Kutta method with different parameters from those given in the text. Use the total number of components in y and the initial condition y(0) = yo as the input to the subprogram. Test the fitness of the parameters by comparing the numerical result from the subprogram and the known exact result for the motion of Earth.Prove:Church’s thesis: The class of computable functions is equal to the class of intuitive computable functions.For each of the following functions, indicate the class (g(n)) the function belongs to. (Use the simplest g(n) possible in your answers.) Please show work to prove the assertions.
- Give a recursive definition of the relation is equal to on N x N using the operator s.Let ∑={a,b} and T be the set of words in ∑* that have an equal number of a’s and b’s. (a) Give a recursive definition for the set T. (b) Show that abbaba is in T by building up from the base case through recursion. (c) Is your recursive definition uniquely determined?is this functions from R to R is a bijection or not f(x)=(x2+2)/(x2+3)