[5] For each of the following functions f find a simple function g such that f(n) = Ⓒ(g(n)). (a) fi(n) (1000) 2" +4". (b) f(n)=n+n logn + √n. (c) fs (n) log(n20) + (log n) ¹0. 100 (d) fa (n) = (0.99)" +n¹
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- Given A = {1,2,3} and B={u,v}, determine. a. A X B b. B X BElectromagnetic Pulse propagating at oblique angle to a dielectric interface Consider a gaussian wave pulse propagating along the z-axis from region 1 with refractive index n1 and onto a dielectric interface y = m z (for all x). To the left of this dielectric interface, the refractive index is n2. Devise an initial value computer algorithm to determine the time evolution of the reflected and transmitted electromagnetic fields for this pulse. e.g., n1 = 1 , n2 = 2 initial profile (t = 0, with z0 < 0) Ex = E0 exp[-a (z-z0)^2] By = n1 * Ex Choose parameters so that the pulse width is at least a fact of 8 less than the z- domain of integration ( -L < z < L). For the slope of the interface, one could choose m = 1.3. Simplify the following Boolean functions, using four-variable Kmaps: (a) F (w, x, y, z) = Σ(1, 4, 5, 6, 12, 14, 15) (b) F (A, B, C, D) = Σ(2, 3, 6, 7, 12, 13, 14) (c) F (w, x, y, z) = Σ(1, 3, 4, 5, 6, 7, 9, 11, 13, 15)
- Show that F(x, y, z) = xy + xz + yz has the value 1 if and only if at least two of the variables x, y, and z havethe value 1.Simplify the following Boolean functions. F(w, x, y, z) = !w!xz + w!xz + !w!x!y!z + !w!xy!z + !wx!y!z + !wxy!z + wxy!z + wx!y!z + w!x!y!z + w!xy!z F(w, x, y, z) = _____answer_______ *Please do not include "F(w, x, y, z) = " in your solution. Add a single space between variables and operators.Consider the function f(x) = 1 3x+1 . We approximate f(x) by the Lagrange interpolating polynomial P₂ (x) at the points xo = 1, x₁ = 1.5 and.x₂ = 2. A bound of the theoretical error of this approximation at x = 1.8 is:
- Simplify the following Boolean functions, using three-variable k-maps: a. F (x,y,z) = Σ(2,3,4,5) b. F (x,y,z) = Σ(0,2,4,6) c. F (x,y,z) = Σ(1,2,3,6,7) d. F (x,y,z) = Σ(1,2,3,5,6,7) e. F (x,y,z) = Σ(3,4,5,6,7)Use the Quine McCluskey algorithm to simplify the function:F(w, x, y, z) = Σ(2, 6, 9, 10, 11, 13, 14, 15)Show all work – no credit will be given for an answer without accompanying work4. Simplify the following Boolean function F, together with the don’t-care conditions d, and then express the simplified function in sum-of-minterms form: a. F(x,y,z) = Σ(0,1,4,5,6) d(x,y,z) = Σ(2,3,7) b. F(A,B,C,D) = Σ(0,6,8,13,14) d(A,B,C,D) = Σ(2,4,10)
- Q_5 Suppose f:RZ where fx=2x-1.If A={x |1x 4}, find f(A).If B={3,4,5,6,7}, find f(B).If C={-9, -8}, find f^ - 1(C)Consider nonnegative integer solutions of the equation x1+x2+x3+x4+x5+x6=30. How many different solutions are there? How many solutions also satisfy: for every i∈{1,2,3,4,5,6}, xi is positive and even?Determine P(A x B) – (A x B) where A = {a} and B = {1, 2}.