Let n ∈ N, prove inductively that
(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then G is Hamiltonian. (b) Let G be a plane graph with n vertices, m edges and f faces. Using Euler's formula, prove that nmf k(G)+ 1 where k(G) is the mumber of connected components of G. (a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then...
Question 2. Monotone Convergence Define a sequence (an) inductively by ai = 1 and an+1 = ("p) (a) Show that, for any k E N, if 0 <a << 2 then 0 < ak+1 <2, and deduce that a, E (0,2) for all E N (b) Show that the sequence (an) is increasing and bounded above. (c) Prove that the sequence converges, and find its limit Question 2. Monotone Convergence Define a sequence (an) inductively by ai = 1 and...
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
5.7. Let n an E C be a multiplicative function defined by a 1 and ifn-2, pθ and 0 < θ < 1 . Prove that, as x → oo, we have where lEpl ano(a) for some constant r. 5.7. Let n an E C be a multiplicative function defined by a 1 and ifn-2, pθ and 0
Let A be an n × n matrix with characteristic polynomial f(t)=(−1)nt n + an−1t n−1 + ··· + a1t + a0. (a) Prove that A is invertible if and only if a0 = 0. (b) Prove that if A is invertible, then A−1 = (−1/a0)[(−1)nAn−1 + an−1An−2 + ··· + a1In]. 324 Chap. 5 Diagonalization (c) Use (b) to compute A−1 for A = ⎛ ⎝ 12 1 02 3 0 0 −1 ⎞ ⎠ . #18 a, b...
(n) 6. Let X ~ Binomial (n,p). Prove that a. Ex=0 (6)p*(1 – p)n-* = ... = 1 b. E[X] = 21-0 x()p*(1 - 2)^-^ = = mp c. Var[X] = x=0x2 (1)p*(1 – p)n-x – (np)2 = ... = np(1 – p) d. My(t) = ... = (pet + 1 - p)n
(1) Let Ω be a set, and let Ao be a family of subsets of $2. Prove that there exists a minimal-algebra in Ω containing 4). In other wo)rds. prove that there exists a 8 σ-algebra A in 12 such that A C A, and . if A, is any σ-algebra in Ω with Ao c A,, then A c A, (1) Let Ω be a set, and let Ao be a family of subsets of $2. Prove that there...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
Prove that trace (A*A)=||A||F2 Definition. Let A be an m x n complex matrix. Its Frobenius noTm, denoted by ||A||F, is defined as follows: m ΣΣΑ Β) / . A|| F = j=1 k=1 1. (7 marks) Let A be as above. Prove that trace(AH A) || A||?