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- 3. (a) Consider the following algorithm. Input: Integers n and a such that n 20 and a > 1. (1) If 0arrow_forwardQ. Let A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5, 6, 7, 8}. How many functions f : A → B(a) ... are injective?(b) ... are not injective?(c) ... are such that f(a) = f(b) = f(c)?(d) ... are such that exactly three elements of A have 8 as an image?(e) ... are surjective?arrow_forwardPlease written by computer sourcearrow_forwardGiven A = {1,2,3} and B={u,v}, determine. a. A X B b. B X Barrow_forwardQ3 In a company, there are several branches. Let us consider a branch of that company having N employees, if the manager is allotted to that branch(s) then he is known to everyone else in that branch. Note that the property of "known to everyone" is unique to a manager. Your task is to find the manager in that branch. Input from the user in main (), the square matrix M where if an element of row i and column j is set to 1 it means that ith person knows jth person. You need to write a function managerId () which returns the id of the manager if present or else returns -1. The function managerId () takes two arguments - the square matrix of N *N and its size N. Call managerId () from the main () output the information about the manager. Assume all diagonal elements to be 1 (as everyone knows him/herself) and there is at most one manager in the company. Example: No of persons N = 3 Input matrix: 0 1 1 1 0 1 Output: manager is person 3 1 0 1arrow_forwardQuestion 3: If every year a tree produces 2 new braches from every existing branch which are at least 2 or more years old, write a function that computes the number of braches of a tree of age k. Assume that at age 0, a tree has 1 branch. For example, at age 1 the tree still contains 1 branch, because the existing branch was not old enough to produce new branches. However, the next year, on age 2, the tree now contains 3 branches: 1 existing branch and 2 new branches. Next, year, the old branch produces two more branches, but the fresh branches only grow themselves. Hence, the total number of branches become 5 on age 3. The following illustration shows this: | -+- | --+-- | -+- \ | / | -+- --+-- -+--+--+- | | | | / | \ --- --- --- --- --- 0(1) 1(1) 2(3) 3(5) 4(11) """ def treeBranch(k):arrow_forwardYou determine a function f(x) that passes through the points (x0, y0), (x1, y1), ..., (xn, yn) with x0 < x1 < x2...< xn. For some other point x * E (x0, xn), you then use f(x *) to approximate the value of the actual smooth function that underlies the data. This procedure is called: O regression O extrapolation O curve fitting O interpolationarrow_forwardLet A = {1, 2,3} and B = {a, b, c, d} What is the function from a to b?arrow_forwardLet f: A → B be a function with A₁, A₂ ≤ A. Determine if the following statement is true or false. Prove it, if you think it's true, or give a counterexample otherwise: ƒ(A₁ N A₂) = ƒ (A₁) Ñ ƒ(A₂). Give an example of a function from N to N that is onto but not one-to-one Let A = {1,2,3, ...,8} and consider the function f: P(A) → N given by f(B) = |B|. Prove or disprove the statement that f is one-to-one.arrow_forward2. Definition: If f(x) is a function, then we say that a value u is a fixed point of f(x) if and only if f (u) = u. Suppose F(x) is a given continuous function and a # 0 is a given real number. Show that u is a zero of F(x) if and only if u is a fixed point of f (x) f (x) = x + a F (x). b. Suppose F '(x) is continuous, u is a zero of F, and F'(u) ± 0. Define f (x) = x + aF(x). Prove there are values of a + 0 and ɛ > 0 so that if uo E (u – E, u + ɛ) and un+1 = f (Un) for n = Hint: Jun+1 – u| = \f (un) – f (u)]. Use the definition of f (x) and the mean а. = 0,1,2, ... then un → u as n → ∞. | value theorem.arrow_forwardH.W2 Minimize the following function using K-Maps: F (A, B, C, D) = Σ m (1, 5, 6, 12, 13, 14) + d (2, 4).arrow_forwardExplain the Wronskian determinant test. Using the Wronskian determinant test, write the program using NumPy to determine whether the functions f(x)=e^(- 3x), g(x)=cos2x and h(x)=sin2x are linearly independent in the range (-∞, + ∞). #UsePythonarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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