) Find U1 A, and n, A, where: a. A = {-i, -i+1, ..., –1,0, 1,.,i– 1,1) for every positive integer i. b. A, = (i,i+ 1, i + 2, ... } for every positive integer i. c. A, = (0,) for every positive integer i. %3D

Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

Q: 1. Show that for all z1, Z2 E C, Then we have: |Z1 + Z2| < |Z1| + |Z2|

A: Using complex properties.

Q: Prove that if a, b e Z – {0}, then gcd(a, b) = gcd(a, -b)

A: Solution:-   gcd a,b= Greatest common divisor of  a  and  b ∗Every integer  that divides  a  and  b…

Q: 5**) (a) Prove that for all positive real numbers n, r, and s, if rs < n, then r < Vn or s < Vn.

A: As per Bartleby's answering policy, we can answer only one question, so kindly post the next…

Q: Prove that I(e) = dr, e> 0.

A:

Q: Let 0 < a < 1 be a real number. Define a₁ = 1+a, Prove that Vn € N, 1 ≤n, an+1 = 1<an< 1 - an + a, -…

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

A:

Q: 5. Let z1 = a1 + b;i and z2 = az + bzi. (a) Find Im[z,Zz + (Z)? – 2,7 ]. (b) Prove that (z)² = z,² .

A: Rational Numbers: All the numbers which can be written in the form of pq. Here, p and q are integers…

Q: Let S = {10n – 1: n e Z}, T be the set of odd integers, and U Prove that S CTNU. {5т + 4: т E Z}.…

A: Consider the given information. Here, S=10n-1:n∈Z And, U=5m+4:m∈Z And, the set T is the set of odd…

Q: (b) Let A¡ = [-i, 1] for every positive integer i. Find U A¡ and N1 Aj.

A: Given that Ai=-i,1 for every positive integer i. That is, A1=-1,1, A2=-2,1, A3=-3,1 and so on. To…

Q: Let  A = (1 1 1, 1 2 2) Find R(A), R(AT), and N(A)

A: Given,A=111122the reduced row echelon form of AMake zeros in column 1 except the entry at row 1,…

Q: Let A € M₂ (R) be given. (a) If AA¹ = 0n, prove that A = On. Does this hold if A € M₂(C)? (b)…

A:

Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

Q: Let U1, U2, ... be i.i.d. and each uniform on [0, 1). For each positive integer n, let q„ be the 1…

A: From the given information, U1, U2,.... are iid and each uniform on [0, 1]. fUi=11-0, 0&lt;Ui&lt;1…

Q: let r ∈ R\ Q. prove that for all a,b,c ∈ Q, a+br=cr iff a=0 and b=c.

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Q: Let A & M and μ*(A) < oo. st B₁, B2,..., BEC, k <co with Bin B „²(AAŬB₂) <« j=1 any two sets E₁ and…

A:

Q: (1 mark) Let x, Y, A, µ E Z. Prove that if 1 = Ar+ µy then I, y are relatively prime.

A: Basic properties of gcd will be used to solve this problem. The typesetted solution in latex is…

Q: Theorem .. ket a and b be non zerointegers- Then there exist Integers r and 8 Such that Hcf (9,b) =…

A: Theorem : Let a and b be nonzero integers. Then there exist integers r and s such that  HCF(a, b) =…

Q: Theorem 1. Let r, y, z be positive integers where r is odd and y is even. Then r, y, z is a…

A: As per Bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

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Q: b) If d = 1, prove that (1 if a is odd gcd(a, 2b) %3D if a is even

A:

Q: find the'mters ection d Exe sy,z]= [a,0,31++ 4,-2,178 [+sy,z]=[6,-12,--11]+5 [1,2,4]?

A:

Q: 2 Q) Let A € M₁ C4) such that A² = A ENA show that det (I+A) = 2

A: If minimal polynomial is totally factorizable that is product of linear factors. Then matrix is…

Q: Let and 001 1 0 0 2 1 A = 0 If the entries of A and v belong to Z3, what is the third entry of Av? 2…

A: Recall matrix multiplication and modulo arithmetic.

Q: (a) Let n €N\{0}, and k e Z. Prove that k-

A: We will use very basic approach for this question in such a manner that it will enter in your head…

Q: 5. Prove that if x, y ER, then x + y ≤ x + y. Use the following cases: (i.) x, y ≥ 0, (ii.) x, y <…

A: We now prove that if x,y ∈ ℝ , then x+ y≤ x+y with the help of given three cases.

Q: 1. Show that for all z1, Z2 E C, Then we have: |Z1+ Z2| < ]Z1] + |z2|

A:

Q: 3. Let n E Z. Give a direct proof that there exist x, y E Z such that n = 55x + 78y.

A: Objective : To prove for any n∈ℤ ∃x,y∈ℤ such that n=55x+78y Prove for n=1, Greatest common divisor…

Q: prove that n E vol(B6)) > II(b: – a;) vol( B jEJ i=1

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Q: Let A = {-i, .,-2, –1,0,1, 2, . i}. Compute the following a) U는1Ai A? = {?} A? = {?}

A:

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: Show that for n, m e Z+, gcd(2" – 1,2m – 1) = 2gcd(n,m) – 1.

A:

Q: Let a,b E R with a # 0, b # 0. Prove that (ab)¯1 = a°1b´1. %3D

A:

Q: o.10. Prove that (a) J,"(x) = ¿[J,(x) – 2J,(x) + J, + glw). n n-2 n' n+2

A:

Q: Let a = 8 and b = 9. Find m,n E Z such that am+bn =1.

A:

Q: b) Prove, by induction, that for any k > 1 and any choice of c1, ... , Ck E R and X1, ... , Xk E V,…

A:

Q: Let m and n be positive integers with gcd(m,n) = 1. Prove that Z/mZ x _Z/ nZ = Z/mnZ

A: Suppose m and n are relatively prime.

Q: 1. For n 2 2, let A = [a E M,(C). Suppose A is diagonally dominant. Show that if af # 0 for all i,…

A: We will use the basic knowledge of matrices and diagonal dominance to answer this question in…

Q: prove for n=l,2,3. e that +2+3+..+h = ncn+l) 2. %3D %3D

A:

Q: 4. Let R be the set of all real numbers, xER and n EN such that n > 2. If (1+x) >0, prove that Pn :…

A: Mathematical induction.

Q: 2. Let A € M, (R) be given. (a) If AA = 0, prove that A = 0₁. Does this hold if A € M₂, (C)? (b)…

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Q: Determine all values of s so that 1 A = | 1 0 1 1 S is nonsingular.

A: Given matrix: A=s101s101s

Q: Let d, a, b E N. Prove that

A: Given,  d, a, b∈ℕ To prove that if d|a+b2  and   d|a-b2, then d|b.

Q: If t,= 4n+1, find t,, t,, t3, and t Express t n+1, in its simplest form. n+1 In-1

A:

Q: 4) Let m,n, r E Z with m # 0 + n. Show that gcd(m, n) divides r if and only if there exist a, b E Z…

A: Let gcd(m,n) divides r. Then we have to prove that, a·m+b·n=r Since, gcd(m,n) divides r than by…

Q: 2. Let A € M, (R) be given. (a) If AA = 0n, prove that A = 0₁. Does this hold if A € M₂, (C)? (b)…

A:

Q: Let m e N and a, b e Z be such that (a, m) = 1 and (b, m) %3D = 1. Let h = 0m(a), k = 0m (b), and…

A: Let m∈ℕ and a, b∈ℤ be such that a, m=1 and b, m=1. Let h=oma, k=omb,  and  n=omab. We need to show…

Q: Find the value of k such that A is singular. 1 0 6 4 -1 0 6 2 k

A: Given. A is a 3X3 matrix. We need to find the value of k for which the matrix is singular. Singular…

Q: Nine Starbucks are located in a 3 mile by 3 mile square area in which there is a 1 mile by 1 mile…

A:

Q: 2. Find U, A, and NE,Ą, where: a. A = {-i, i} for every positive integer i. b. A = {i°, i', i²} for…

A: