Prove that lim f dm = 0 (note that the set ([-k, k]d)e is the complement of the cube [-k, k]d).

Q: We denote the set of polynomials with real coefficients as R[X] := {ao + a₁X + a₂X² + ... + a₂X": n…

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Q: Let a relation S be defined on A = {0, 1, 2, 3} as follows. S= {(0, 0), (0, 3), (1, 0), (1, 2), (2,…

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Q: b) Find the domacn of f a) Evaluate f(2,-1,6). c) Find the tange of f.

A: Here, the given function is: f(x,y,z)=ez-x2-y2...(1)

Q: S ⊂ R3 be the set consisting of 2 points (1, 2, 2) and (1, 2, 3). Find the annihilator of S. Show…

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Q: D) Find 3 linear e) Find different combination that is, generates Set T Span (T) = find the of a…

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Q: How would you find an isomorphism from f: Z3 X Z14 -> Z21 X Z2. Kindly elborate.

A: Given That : A mapping f :Z3×Z14→Z21×Z2 To Find : The isomorphisms from f :Z3×Z14→Z21×Z2.

Q: consider the space generated with the set of functions  {x, x^2} , show that the set of elements of…

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Q: Let N R R be any norm on Rn. Show that N is a convex function (give all the details for the proof).

A: Given N :R^n→R be a norm on R^n.      To show : N is a convex.     Definition of norm :…

Q: 11) Let H < K < G. Prove that |G:H|= |G:K| · |K:H| (Do not assume G is finite).

A: Given : H≤K≤G We have to prove that G : H=G : K. K : H

Q: In the frieze group F7, show that zxz = x-1.

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Q: 2. Prove that ƒ : (R × R, +) → (R, +) given by f(x, y) = x + y is a homomorphism. Is g : (R × R, +)…

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Q: Prove the Chinese Remainder Theorem below: Chinese Remainder Theorem (CRT): Let r, m1, . . . , mr ∈…

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Q: 1. Let V be an inner product space. Suppose f,g E V such that f -gg (a) Prove that 1I2 = |lf- gll2 +…

A: To prove the required norm property in an inner product space, under the given conditions

Q: Consider the relation S on N defined by uSy⟺5|(y+u). Is it reflexive?

A: No it is not a reflexive relation

Q: 1 0 1 ~EHEMMEHO V 4 2 0 Let v = 1 0 4 O C.S O d. {v, - and S= { v₁ O a. (v. v₂} 1 2 9 O b. {V₁, V 3…

A: Since you have posted multiple questions, we will provide the solution only to the first question as…

Q: Prove: If ƒ e R(a) on [a,b] and g e R(a) on [a, b] , then fg e R(a) on [a,b]

A: Given if f∈Rα on a,b and g∈Rα on a,b. We have to prove that fg∈Rα on a,b. Since, f∈Rα on a,b then f…

Q: Prove that Z[i] is a Euclidean domain.

A: Theorem: Proof:

Q: Let H = Span{1+x,1+x², 3+x +2x², 2 + x + x²}. what is dimH?

A: Solution

Q: Claim that = (0.0.0) 2 a) show √ E OF √₂, √₂. b) Show OF J₁ () show J₁ s={√₁₁ √√₂, √√₂} S= √√₂₁ are…

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Q: B: Let T be the projection on W₁ along W₂ b) Prove that W₂ is the Nullspace of T.

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Q: Suppose that A₁, A2, and A3 are distinct non-collinear points in a plane. Define an infinite…

A: Given : A1 , A2 and A3 are distinct non-collinear pints in a plane. An ∀ n≥4 be the reflection of An…

Q: Consider the set of matrices 0 S = {(a+b):abER} a, bЄR

A: the given functions \(f\) and \(g\) are homomorphisms of rings, we need to verify if they preserve…

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.