Define T R → R by T (a, b, c) = (a – c, b+ c) Is T one to one? [ Type Y for yes and N for no. Is T onto? Type Y for yes and N for no.

Q: Let w = xydx + 2xdy - ydz E 2'(R) and let a: R a(u, v) = (uv, u², 3u + v). Compute do, a*o, d(a*o),…

A: Given ω=xy dx+ 2x dy - y dz in Ω1ℝ3x,y,z And let α:ℝu,v2→ℝx,y,z3 be given by  αu,v=uv, u2, 3u+v. To…

Q: Consider the following inner product R?. on ((T1. Yı), (T2. Y2)) = ax122 + by,Y2 If ||(1, 3)||² =…

A: We will find out the required value of a+b.

Q: Let {X;} be i.i.d.~ Exp(0) and a; > 0. Denote by Z = min{a,X1,..., amX,}. Suppose that 0. Prove that…

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Q: 69. Let F(x, y) = (x,0). Prove that if C is any path from (a, b) to (c, d), then F. dr (o2

A: Given: F(x, y) = <x, 0>. C is any path from (a, b) to (c, d).

Q: Let G = Z31, S = {1,5, 6, 25, 26, 30}. %3D %3D Prove that Cay(G, S') is normal.

A: It is given that

Q: Let f: R R given by f(z, 9) = (x+2y, z- y). Of is 1-1 of is not 1-1 of is not onto of is onto

A: Given that fx,y=x+2y,x-y

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: Let 2= {p₁₁₁r, s, t, u,v} and d = {{p.a.v}, {st.v}-{4.0}} q, U, V 1 1 field containing d. construct…

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Q: Let n E w. Suppose that f:n → A is onto A. Prove that A is finite.

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

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Q: Let E={(r,0,z) : os es T, OS zs 1, 5Vzs rs 5. Compute

A: Introduction: The triple integral ∫∫∫Ef(x,y,z)dV with the region E=(x,y,z):a≤x≤b, c≤y≤d, e≤z≤f can…

Q: Prove that r(z) = (In-)²-'dt , Re(z) > 0.

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Q: Let S and T be subset of R. Prove the following: cl(SUT) = (cl S) U (cl T)

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Q: Q1. Let T={x,y,z}. L={Ø,x},{x,y},T} is a o-algebra over T. Prove that: (a) (b) (T,L) is a measurable…

A: (a) Given that T=x,y,z, L=∅,x,x,y,T is a σ-algebra over T. Since every sigma algebra is measurable.…

Q: Let (W, t > 0) be standard BM. Show that P(n|w,1 – Wa sc) → as n → 0.

A: From the given information,  (Wt, t≥0) be standard Brownian Motion. Note: In the given situation,…

Q: Let T : V → U be a homomorphism, then Ker T = { 0 } iff T is one-one

A: A linear transformation is said to be injective or one to one if provided that for all v1 and v2 in…

Q: A function from ZxZ to Z defined by f(m, n) = m-n is an onto function. O True O False

A: The given function is                                     f: ℤ×ℤ→ℤ  such that    f(m,n)=m-n. A…

Q: f:A→Z is not onto and f:A M is onto.

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Q: If S is a subset of T , then u* (T) su* (S).

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Q: The orthogonal projection of y onto u is the same as the orthogonal projection of y onto cu whenever…

A: solution

Q: The Wronskian of y,(t)=e 2, y,()=e" is-e St. Select one: True False

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Q: Let E Os 0s n, 0s zs 1, 8Vzs rS 8. Compute

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Q: 10. Let X , j= 1,.., n, be i.i.d. r.v.'s such that E(X,)= µ, o'(X)=o', both finite. j' Show that E(X…

A: The normal distribution: A continuous random variable X is said to follow normal distribution if the…

Q: Qt Let T= {carb) a,b ER\fopp and of defined onI by Carb)*(c,d)= (ac, bd) for all (a,b), (c,d ET,…

A: The set T is defined as, T=a, b | a, b∈ℝ\ 0 The operation * is defined on T as follows, a, b*c,…

Q: Let T={a: a ER}, then T is Borel algebra.

A: Borel sets of real numbers are definable as follows. Given some set, S, a σ-algebra over S is a…

Q: Prove that (aA + bB)t = aAt + bB t  for any A, B belongs to Mm× n(F) and any a, b belongs to F.

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Q: Let v e R" and letk e R. Prove that S = {x € R" :x•v = k} is an affine subset of R". %3D

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Q: et S be a nonempty subset of R and let k E R. Define kS {ks : s E S}. Prove if k > 0, en sup(kS) = k…

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Q: Let T={[a, ) : a€R}, then T is not Borel algebra.

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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: Let f(z, y, z) = z2 +y? + z²,z = /utv, y = /u – v, and z= uv, Find in terms of u and v.

A: we have to partial derivative with respect to u by using chain rule

Q: Let T={[a,0) : aER}, then T is not Borel algebra

A: Given a statement

Q: 2. Let S,T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly…

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Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: When is S \ T = T \ S for sets S, T?

A: Given that two sets S and T. The definition of S\T=a∈S,a∉T and T\S=b∈T,b∉S. Here, neither of these…

Q: Let h: (Z, +) (R – {0},.) define as h(n) = n² then h is 1-1 O 1-1 and onto onto No one O O O

A: If a horizontal line can intersect the graph of the function only a single time, then the function…

Q: Let P = {(z, y, 2) E R° | az + by + cz = 0} where (a, b, c) # (0,0,0). (a) Prove that (a, b, c) ¢ P.…

A: Given that P=x,y,z∈ℝ3 : ax+by+cz=0 where a,b,c≠0,0,0 (a) The objective is to prove that a,b,c∉P Let,…

Q: Consider f : N → N, g : N → N and          h : N → R defined as f(x) = 2x,g (y) = 3y + 4 and h (z) =…

A: Given, Consider f : N → N, g : N → N and          h : N → R defined as f(x) = 2x, g (y) = 3y + 4 and…

Q: If S is a subset of T, then u*(T) su* (S)

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Q: Let z, w e C. Clearly prove that z+w = z + w.

A: Since you have asked multiple question , we will solve the first question for you. If you want any…

Q: Consider Uu=4uzzy u(x,0) = u(x), u₁(x,0) = u₁(x), on the domain -∞<x<∞, 0≤t<∞o. Suppose that both…

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Q: Let T={(a, o): aER}, then T is Borel algebra

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Q: Consider R? and R3 with the Euclidean (c) norms. Compute || T || for T : R2 →R3 defined by T (x1,…

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Q: Let an 1+1/V2+1/V3+...+1//n , Prove {an} is unbounded. %3D

A: Given that, an=1+12+13+...+1n

Q: Please solve...

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Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

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Q: Let x(v, w) = } In(v² + w²). Show that = 0

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Q: IIC sUUset of R". tve R" and let k e R. Prove that S = {x e R":x.v = k} an affine subset of R". %3D

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Q: Prove that S[x] = {a+bx where a,b in R} is subring ℝ[x].

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Q: If ToS: V Z is onto, then T is onto.

A: