Suppose that f(i) is a periodic function of period 2a with
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- Suppose that the equation ax b .mod n/ is solvable (that is, d j b, whered D gcd.a; n/) and that x0 is any solution to this equation. Then, this equation has exactly d distinct solutions, modulo n, given by xi D x0 C i.n=d / fori D 0; 1; : : : ; d 1arrow_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_forwardLet A = {1, 2, 3, 4} and B = {a, b, c}. Give an example of a function f: A -> B that is neither injective nor surjective.arrow_forward
- 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_forwardA function is said to be if and only if f(a) = f(b) implies that a = b for all a %3D and b in the domain of f. O a. No answer is correct O b. One to One Oc. One to many O d. Many to onearrow_forward3. (a) Consider the following algorithm. Input: Integers n and a such that n 20 and a > 1. (1) If 0arrow_forwardA service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation. p(x, y) 0 1 2 0 0.10 0.05 0.01 x 1 0.06 0.20 0.07 2 0.06 0.14 0.31 (a) What is P(X = 1 and Y = 1)? P(X = 1 and Y = 1) = | (b) Compute P(X ≤ 1 and Y < 1). P(X ≤1 and Y < 1) = (c) Give a word description of the event { X0 and Y + 0 }. One hose is in use on one island. One hose is in use on both islands. At least one hose is in use at both islands. At most one hose is in use at both islands. Compute the probability of this event. P(X +0 and Y 0) = [ (d) Compute the marginal pmf of X. x 0 Px(x) Compute the marginal pmf of Y. y Py(y) 0 Using Px(x), what is P(X ≤ 1)? P(X ≤ 1) = = 1 2 1 2arrow_forwardLet X = {1,2,..., 100} , and consider two functions f: X → R and g : X → R. The Chebyshev metric of f and g is given by: d(f, g) = max |f(x) – g(x)| Write a functiond (f,g) that calculates the Chebyshev metric of any two functions f and g over the values in X.arrow_forwardDefine a function make_derivative that returns a function: the derivative of a function f. Assuming that f is a single- variable mathematical function, its derivative will also be a single-variable function. When called with a number a, derivative will estimate the slope of f at point (a, f(a)). the Recall that the formula for finding the derivative of f at point a is: f'(a) = lim h→0 where h approaches 0. We will approximate the derivative by choosing a very small value for h. The closer h is to 0, the better the estimate of the derivative will be. def make_derivative (f): """Returns a function that approximates the derivative of f. >>> def square (x): f(a+h)-f(a) h Recall that f'(a) = (f(a + h) - f(a)) / h as h approaches 0. We will approximate the derivative by choosing a very small value for h. # equivalent to: square = lambda x: x*X return x*x >>> derivative = make_derivative (square) >>> result = derivative (3) >>> round (result, 3) # approximately 2*3 6.0 |||||| h=0.00001 "***…arrow_forwardGiven g = {(1,c),(2,a),(3,d)}, a function from X = {1,2,3} to Y = {a,b,c,d}, and f = {(a,r),(b,p),(c,δ),(d,r)}, a function from Y to Z = {p, β, r, δ}, write f o g as a set of ordered pairs.arrow_forwardLinear Dynamical system help, PLS more detals and clearly to see.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
- Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole