Q2. (a) Prove that if ), and >. are segma algebras over X, then 2n2, is a sigma algebra over X. (b) Is ΣΣ a sigma algebra over X? Prove your answer.

Q: A Lie algebra L is said to be reductive if Z(L) = Rad(L). (For parts (b), (c), assume chark = 0 andk…

A: a) Let L be a reductive lie algebra and let L=I1⊕I2⊕...⊕In be a decomposition of L into simple…

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: Now just as with Example 2.16, prove that if f, g, and three polynomials in R(r], then (f+ g) + h =…

A: Introduction: In mathematics, functions are the most fundamental aspect of calculus. Functions are…

Q: J1 Q2. (a) Prove that ifE, and >, then nE, is a sigma algebra over X. (b) Is ,u, a sigma algebra…

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Q: Prove that 1¹ 1². If: T: (1², |·|₂) → (¹', |· | T: (1²², ·|₂) → (¹², ||·| is also compact. is a…

A: Definition: l1: The space l1 is the set of all sequences x=xn of complex numbers such that the sum…

Q: Let A be an associative algebra. (a) Show that for any a E A the map da : A A defined by da(b) = ab…

A: Let A be an associated algebra.  To prove that for any a∈A, the map da:A→A defined by dab=ab-ba is a…

Q: Let A= 4 4 2 1-1 (a) Compute det(A) using the cofactor expansion along the third row. (b) Is A…

A: The adjoint of a square matrix A = [aij]n×n is defined as the transpose of the matrix [Aij]n×n ,…

Q: A) Let f: Z→ N as follows: 2n – 1 for n > 0, - f(n) = -2n for n < 0. Show that f is a bijection.…

A: Bijective function

Q: Q[x]/ is I.D not field O Not I.D O Field

A:

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: Exercise 3.9 Suppose X is the set of real numbers, B is the Borel 0-algebra, and m and n are two…

A: Given X=ℝ, the set of all real numbers. The a Borel σ-algebra, B,  is a  σ-algebra generated by…

Q: Let Z* be the set of positive integers. We define the binary operation f:Z* x Z* → Q by f(a, b) = .…

A: The binary operation is defined from the cross-product of a set to another set. The binary operation…

Q: Let k≥n. Give a combinatorial proof of the identity k Σ(*) (*) = (¹ + *). n r=0

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Q: Let g(x) and h(x) belong to Z[x] and let h(x) be monic. If h(x) divides g(x) in Q[x], show that h(x)…

A: Given: Let's take g(x) and h(x) belongs to Z(x) and take h(x) as monic.

Q: Let S = {( ) laeR). Then S is a Field O True False

A: I have proved the all conditions for field.

Q: rove the following identities in a Kleene algebra ((a*b)* ba*)* = ( (b + (a + b)*bb)a*)* the…

A: To Prove ((a*b)* ba*)* = ( (b + (a + b)*bb)a*)* Take RHS ( (b + (a + b)*bb)a*)*          ( (b…

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: Find the g.c.d. in Z [i] of 2 and 3+ 5i.

A:

Q: (i) If A E F and BE F then ANBE F. (ii) If A and B are subsets of N such that ANBE F then either A E…

A:

Q: (a) Let X = {m, n, p, q}, and = {ø, X, {m}, {n}, {p, q}, {m, n}, {n, p, q}, {m,p, q}}. Is a sigma…

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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: a We have shown that C = b a the isomorphism f: CC where f - | a, b € a, beR is field. Show that CC…

A:

Q: Theorem: if I is a Positive implicative Bck-algebra Then (Lx),,Lo) is a BCK-algebra

A: Consider the given theorem, '' if X is positive implicative Bck-algebra then Lx,⊙,L0 is a…

Q: Let A = (a, a, as) and B = {b₁,b₂, b3} be bases for V, and let P = {[b₁]A, [b₂]A, [b3]A} (are)…

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Q: 6. Show that Q(V3) = {a + bv3 : a, b e Q} is a field.

A:

Q: Let -{[:);}--{:)[:} -{{)}} Both B and C are bases for R2. If {x}s =: -[:], what is {x}c?

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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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Q: *).). (a) Show that X is measurable with respect to o-field G if and only if o(X) C G. Show that X…

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Q: 8 6. Let A=|3 5 7 | 4 9 (i) Evaluate det(A) by cofactor expansion. 2 (ii) If A is invertible, find…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.