Let K be the collection of all compact sets in R and Cx the collection of all countable subsets of K. For the Borel o-algebra on R, prove that B(R)= Uo(C), CECK where o(C) denotes the smallest o-algebra on R containing C.

Q: T: M2x2 (R) → R2 [x] is given by r(; ;) a С 3 (а +ь+с+d) + (а+ 2b + 3с + 4d)ӕ + (а + 36 + 5с + 7d)…

A: To solve this problem, we use the property  of linear transformation and to check condition which is…

Q: Explain why the operation (A, B)- tr(AB) defines an inner product on S, but not on Mn.

A: We know that M22 is the vector space of all 2×2 matrices with real entries and S2 is the space of…

Q: Prove that there is no homomorphism from A4 onto Z2.

A: Fundamental theorem of homomorphism: If f:G→H is homomorphism between two groups, then there exists…

Q: 5. Show that M:= {ECR: E is countable or E is countable} is a o-algebra.

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Q: 3- For any R-module ,let M [x]denote the set of all polynomials in x with coefficient in M, that is…

A: Sol

Q: (a) Pieve that f 2 and Ź are segma algebras s over X th

A: Definition of Sigma Algebra: A Collection ∑1  of subsets of an arbitrary algebra if : (1) X ∈∑1…

Q: How do you prove that for any m ∈ ℤ the product of any a,b ∈ ℤ[√m] belongs to ℤ[√m]. ? (where ℤ…

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Q: prove That Lemma: Let A be a fuzzy ideal of Roah and br be fuzzy singleton. Then (a+b)noA-ahobno A

A: Given Equation:(A∘B)H∘A=AH∘BH∘A

Q: Let L be a finite dimensional Lie algebra. Any finite dimensional representation of L is completely…

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Q: Let G = {[1 0 0 1] ,[−1 0 0 1] ,[1 0 0 −1], [−1 0 0 −1]} . Is G ∼= K4? If yes, give an explicit…

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Q: Q3. Write the Disjunctive normal form of Boolean Polynomial p = x (y+z)' + ( x y + z ) x'.

A: Disjunctive Normal form: A Boolean expression is said to be disjunctive normal form (DNF) if it is…

Q: . Show that ||(1, 2)||1₁ = |x1|+|x₂| is a norm on R², but that it does not satisfy the parallelogram…

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Q: Show that for any two finite-dimensional L-representations V and W of L: (a) Hom (V, k) ~ V* as…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: Prove that no element of ℤ/n is both a zero divisor and a unit.

A: To Determine :-  Prove that no element of ℤn is both a zero divisor and a unit.

Q: is an R-module homomorphism snomorphism if and only if k

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Q: The class of all finite disjoint unions of intervals of the form (a, b), (-0o, b), (b, o0), 0, and R…

A: Given: The class of all finite disjoint union of intervals of the form (a,b] ,(-∞,b] ,(b ,∞) ,∅ and…

Q: Let T: R² R² be given by T(x1, x₂) = -[at] It is known that I is an isomorphism. T-¹ is given by the…

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Q: if T is an operator in a real vectorial space with characteristic polynomial (x²-2x+5)². What are…

A: For the solution follow the next steps.

Q: et FCK be a field extension and let R be the alg K. Then R is a subfield of K and FCR.

A: let alpha and beta are in R

Q: Prove that π2 -1 is algebraic over Q(π3).

A: We have to prove that π2-1 is algebraic over Qπ3.

Q: Z[] {a + bi| a, b EZ}. Show that Z[] Let denote the set a subring of C.

A: Here to show z [i] is a subring we need to show show z [i] is closed under substraction and…

Q: Theorem Let A a o – algebra. Let P:A (0, 1] so that P(2) = 1 and it is additive. Then it is…

A: Given that A be a σ-algebra. Let P:A→0,1 so that PΩ=1 and it is additive. To prove that given parts…

Q: 3- Let a: (F(R), +..) → (R, +..) be the map defined by a(f) = f(3)-f(0). Then o is a ring…

A: Given: σ:Fℝ, +, ·→ℝ, +, · be the mapping defined by, σf=f(3)-f(0) To determine: Whether σ is a ring…

Q: Suppose R is commutative. Prove that the map oa : R[x] → R where $₁(p(x)) = p(a) for some a € R is a…

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Q: 2. Suppose someone is trying to illustrate Cantor's diagonalization proof that |X # |P(X)|, where X…

A: A bijectIon means a mapping which is both one - one and onto .

Q: Prove the following identities in a Kleene algebra. (a + b + bb)* = (a + b)*

A: to prove  a+b+bb*=a+b* Explanation L.H.S a+b+bb*=a*b+bba**                   =a*b+bba**…

Q: Let M, and M2 be two R-modules and N, and N2 be two submodules of M, and M2 respectively. Also, let…

A: Here we need just one fact that if f:X→Y is a function defined from X to Y and if y∈Y, then f-1(y)∈X…

Q: Let N/ Q be a normal extension with N ¢ R. Prove that there exists some element TE Gal(N/Q) such…

A: Let , N/ℚ be a normal extension with N⊄ℝ. and Let  ζ : N→N  defined by ζα=α→ (complex conjugate of…

Q: Every submodule of a noetherian (artinian) module is noetherian (artinian).

A: Every submodule of a noetherian (artinian) module is noetherian (artinian).

Q: Q18 (6) Let a,b, c be three R. elements. and (ab) = 1.g abc then Show that a/c- (b) Let A an d B be…

A: PART A Given that , If a, b, c be the statements of a Euclidean ring R and gcda,b=1 . If a|bc,  Then…

Q: Prove that S = {p(x) = R[x] | p(2) = 0 and p(3) = 0} is a subring of R[x].

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Q: Construct a nontrivial homomorphism $: Zg → Zı5 >

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Q: 2x − 3y = −1 4x + 3z = 4 2y3z = -2

A: Given equation can be written as: 2x-3y+0z+1=04x+0y+3z-4=00y+2y-3z+2=0 Charmer's Rule to find x, y,…

Q: Consider ℚ(√3) = {a + b√3 | a, b ∈ ℚ}. (a) Show that ℚ(√3) is a subring of R. (b) Show that the…

A: Ans- (a) To show that ℚ(√3) is a subring of R, we need to show that it is closed under addition,…

Q: (3) Show that (C SRICs countable or C is antable) is a -algebra on R.

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Q: Let M be an R-module, and let N be an R-submodule  of M. Then M is noetherian (artinian) if and only…

A: Given:M is an R-module, and N is an R-submodule of M. To Prove:M is Noetherian (Artinian) if and…

Q: Solve the initial value problem x'(t)=Ax(t) for t20, with x(0)=(5,4). Classify the nature of the…

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Q: 3- For any R-module ,let M [x]denote the set of all polynomials in x with coefficient in M, that is…

A: Given: px=m0+m1x+...+mrxrq(x)=n0+n1x+...+nsxs To find: Mx≅Rx⊗RM

Q: Let K = ℚ(sqrt2, i ). Find all isomorphisms of K onto a subfield ℚ(bar) as extensions of the…

A: Let K=Q2,i Let ϕ:K→K, So ϕ depends on Q2 and Qi. Now, minimal polynomial of 2 is x2−2 So, other…

Q: Pr(A) := lim n→∞ |An [1, n]| n exists and Pr(A) = {0, 1] that that F is an algebra (which is to say…

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