Prove that on average, permutations of length n have H, cycles, where Σ Hn i=1

Q: Given the bases B = {u,,u2} and B'={u'1, u'2} for R², where and u'1= The transition mat rix Pg-8'…

A: Given bases  are B=u1,u2 and B'=u1',u2'  where u1=10,u2=01 and u1'=12,u2'=11 To find the transition…

Q: De Morgan's law

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Q: Let (A, 3) be a poset, and let BC A. Prove that B has at most one greatest lower bound.

A: Let, A,<~ be a poset and B is a subset of A and a∈A. Then by the definition, a is the greatest…

Q: For two R-modules Mand N Show that Hom (M,N) any with the composition is not agroup?

A: Given that M and N are two R-modules. We have to show that HomM,N is not a group under composition.…

Q: Let n, k, m EN with n2k21. prove that

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Q: Consider the Venn diagram below. Illustrate the other De Morgan's law by selecting the region(s)…

A: GivenAc ∪ Bc

Q: If sup(S) exists, then it is unique ylgn ihi

A: The solution is given as

Q: Prove the following version of Darboux's Th

A: To prove the Darboux's theorem:

Q: Let p». Prove that, for every e > 0, a.a.s. the largest component of G(n, p) has size at least…

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Q: Prove that the following relationship is true: nCr + nCr + 1 = n + 1Cr + 1

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Q: A The legih in pnefers of the fout sider f a building angx Cox- 5-9*, and Axt a. Calculate The…

A: Solution :   (a)   Total length around the building               = 3x + (6x - 5) + 9x + (4x + 1)…

Q: 7. In ABCD, AC and BD divide each other in half at E. (a) Prove that ZDBC ZBDA. (b) Prove that ADBC…

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Q: Let a e Z. Prove that if 6 | a and 10 | a, then 15 | a.

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Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: Show that A is skew symmetric if and only if ai j = −aji for all i, j.

A: Suppose A be any square matrix of order n such that A=a11a12⋯⋯a1na21a22⋯⋯a2n⋯⋯⋯⋯⋯an1an2⋯⋯ann First,…

Q: 5. Prove that if A and B are similar then dim(ker(A)) = dim(ker(B)).

A: We have to prove dim(ker(A)) = dim(ker(B)).

Q: 3 a A- E Prove ZBED = ZABC + ZCDE. %3D

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Q: 2, Say whether the Runction is even, odd, or neither. Give reasanS for your dnswer. ga) = x"+ x*-1

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Q: Given: MN || PO, MP || NO Prove: MP = ON

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Q: Let c E R and suppose that an → a. Show that can → ca.

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Q: Let a and b belong to Sn. Prove that ßαß-1 and a are both even orboth odd.

A: Let, a and b belong to Sn.   Consider α is the product of a 2-cycles, and β is the product of b…

Q: Given: mAB = x Prove: mz1 180° A B

A: Now let O be the center of the circle. Join A and B with the center, we get

Q: In the domain of Gaussian integers Zli), the element 19 is * O Reducible O Unit Irreducible

A: Answer is irreducible. Proof is given below.

Q: Show that if A is nonsingular and AB = O for an n × nmatrix B , then B = O .

A: We have given  matrix A is non singular and AB=0 for an n×n matrix  B. We have to prove that B=O

Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

Q: Prove that if n 2 3, the only element o of S, satisfying oy =ya for all y E S, is the identity…

A: Let n≥3. Let σ∈Sn such that σγ=γσ for all γ∈Sn. Suppose σ≠i, the identity Then there exists…

Q: Let Sn=Vn. Find an and show that an O but that Sn=ER=1@k diverges. ->

A: Solution:-

Q: Show that every positive sol fron of Ay + B, k=osl,2- & bosunded, %3D JK is i-e, ogy sB-if A<1. 1-A

A: Given yk+1=Ayk+B, k=0,1,2,... Then the objective is to find the solution at which the sequence is…

Q: c If in is an involution of X, show that (r, y)in = (yi", rin) is an involution of X12). %3D

A: Given X is a lattice. Given in  is an involution of X To show: x,yin=yin,xin is an involution of X2…

Q: Give three examples of Bernoulli rv's (other than those in the text). (Select all that apply.) Ox =…

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Q: eb Let A E M₁ (TR) be given If AAt = On HOME JATOT prove that A = On - Does this hold if A EMn (C)?

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Q: (2) Show that if dim(V) = k (so in particular, T has k distinct cigenvalues), then there exists v e…

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Q: Find an explicit formula for s, if so = 1 and %3D Sn =2+2s0 + 2s1 +... +2s, 2+ Sn-1, Vn >1.

A: Generating Functions for a sequence a0, a1,a2,………an,……………… is an infinite series of the form ,…

Q: In R* whire n eN, let こ and イ.ES":(tx € E),(ヨr >o)ud tlnt Br (x) CE lang the deginition 4 topelogs…

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Q: Prove by Contradiction that if    a 0, which implies  a< = b

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Q: Prove the following in Z[i]: if a y and 3|y with a and B relatively prime then aß | y. in

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Q: Suppose a E Z. Prove that for any n E N, 5|2"a implies 5|a.

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Q: Given: mAB = x Prove: mz1 = 180° - x A D Exercises 25, 26

A: Given:           m AB⏜ = x   To prove:              m&lt;1 = 1800 - x

Q: Is ZXZ cyclic? [Here Z means (Z, +).]

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Q: Find the representation B of L with respect to S' and T'.

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Q: In ALMN, LN is extended through point N to point 0, mZMNO = (5x – 13)°- mZNLM = (x – 4)°, and m/LMN…

A: Solution-

Q: Prove for all o, TE Sn that oro-1 € An. TTO

A: Permutation

Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: When A has full rank, the pseudo-inver en m≥n, and as A+ = AT (AAT)-¹ when n A = I, so A+ is a left…

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Q: Find the a E Z935 such that it satisfies the following system of congruence: I = 9 (mod 17) I = 1…

A: We will solve it by Chinese Remainder Theorem The general congruence relation is of the form x = bi…

Q: Let SCR and let c e R be a cluster point of S. Prove that for each e >0 the set W := Sn(c-e,c+e) has…

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Q: Show that if E,=1 a, diverges, then En=1 | |a,| diverges. %3D

A: Given, We want to prove that

Q: Assume C and D are chosen at random from {1,2...,n}. Let pn be the probability that C + D is a…

A: The number of pairs (c,d)  with sum m2 is m2-1 for m2≤n and 2n-m2+1 for n&lt;m2&lt;2n. Hence,…

Q: Question 2: Show that the set S = {rVT + V2:rE Q) is dense in IR. Explain and show your work.

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Q: Hw/O Find Adi A if A. O kind Adj A il A=

A: (1). The given matrix is:  A=521  314  463