By considering u*u* N * N, prove that 00 Σ01(d) 01 (1) = Σ doo(d) oo (7) · d\n d\n
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- n 3. (a) Prove that Σ³ = = n²(n+1) ². j=1 (b) Evaluate fr³ dx directly from the definition, without using the Fundamen- tal Theorem of the Calculus. (Hint: use Lemma 4.13 or Theorem 4.16.)OY-(A'+C ). (B+ D') QUESTION 5 Boolean Algebra X (X + Y ) is equal to. O X. Y O X+Y O X.X'+X.Y O X.X' QUESTION 6 Using the following Karnaugh, Find the simpProve that funcdiua u = ay =2X - 2xy -y'+x* is har monic . Final - funclion J such thuf fas = u+iV is andlytic
- the book is "First Course on Fuzzy Theory and Applications"3. (.) Denote the number of derangements of {1, 2,..., n} by Dn. Using the formula for the number of the derangement that we found in class, provide a proof of the relation D, = (n – 1)(D-2 + Dn-1); (n = 3,4, 5, ...).Deline two opero tions + nd n A=Zx dis follaus for la,0) ond (Cd) (a,b)+ (c,d) = (ate, btd) ond (9,6)(c,dl- (ac- bd, bc tad) Show that for every Ca,b) nd (e,) in A there exist Co,lond (rs) in A Such the t (a,bl = ((P, q) * (aidl) + (ris) R4s?18. Let T) Σ-1() k=0 (a) Use Pascal's Triangle to compute T(n) for n = 1, 2, 3, 4. (b) Prove that T(n) = 0 for all n > 1. Hint: Expand (a + b)" and evaluate at a = 1, b = -1. %3D(15) Let G = Z/mZ, let d' be a divisor of m, and let d= m/d'. Prove that the congruence class d'+mZ has order d.Let P(Z) = 0.43, P(Y) = 0.40, and P(ZnY) = 0.18. Use a Venn diagram to find (a) P (Z'NY'), (b) P (Z'UY'), (c) P (Z'UY), and (d) P (ZnY'). (a) P(Z'NY') = (Type an integer or a decimal.) C Y OWhy is inversion [that is, Φ(x, y) = (-x, -y)] not listed as one ofthe four kinds of isometries in R2?5. Let U U[5, 10] and let X I15.7) (U) be the indicator function of [5, 7) and let Xn I15,7+1/n?)(U). Prove that X, + X with probability one.d)) A random process is defined as X(t) = μ + N(t), where N(t) is white noise with auto covariance function Cxx (t₁, t₂) = 0(t₁)8(t₁- t₂), where Ø(t) is abounded function of t and 8 is the unit impulse function, prove that {X(t)} is a mean ergodic process. DART D 2035SEE MORE QUESTIONS