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

A: To show that for any tow finite-dimensional L-representation V and W of L HomkV, W≃V*⊗W as L-module

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: Show that every nonzero element of Zn is a unit or a zero-divisor

A: The elements of Zn are 0, 1, 2, …, n-1. The non zero elements of Zn are 1, 2, …, n-1. We know that…

Q: Prove that [x] ∈ ℤ/n is a zero divisor if and only if there exists an integer y >1 such that y | x…

A: Let x∈ℤn is a zero divisor. There there exists a nonzero element y such that  xy=0 in ℤn. Hence xy…

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: State and prove Euclid’s Lemma.(you may assume without proof that for integers x and y there exist…

A: If x and y are integer not both zero then there exist integer k and l such that kx+ly=gcdx,y

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: (Z × Z)/{(4, 7)) ~ Z

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Q: Using the natural homomorphism(which takes every integer to its equivalence class) of Z onto Zk for…

A: The idea is to reduce the equation mod 5

Q: Find all the homomorphism of (z,+) onto (z,+)

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Q: Give an example of each of the following, and show that your example satisfies the "highlighted"…

A: For the solution follow the next steps.

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, for all constructible numbers y € R, x³ – 2 is irreducible over • Q(r).

A:

Q: Prove that the degree of any irreducible factor of x 8 - x over Z2 is1 or 3.

A: We have given that, fx=x8-x We need to show that degree of any irreducible factor of fx over ℤ2 is 1…

Q: De ne function f : Z ! N by f(n) = 2n if n > 0 and f(n) = 1 ? 2n if n   0. (a) Observe that the…

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Q: SHOW THAT U IS TYPE ITS MINIMAL THE FORM cao, a₁ A ₂₁. ALGEBRAIC OVER Q AND POLYNOMIAL BELOW IN…

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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: Solve using method of undetermined coefficients: y" + y' - 2y = x². %3D

A: Solution :-

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: 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 T be an operator on an innner product space V, and suppose that ||T(x)|| = ||x|| for all x in V.…

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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: 3. Show that Z, -

A: