Prove: [a] is an element of Zn has a multiplicative inverse if and only if gcd(a,n)=

Q: Q. If E, and E, are segma algebras over X, Σ EUL, is a sigma algebra over X а. '2 b. n2, is a sigma…

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Q: In the ring Z[x]/ 1- The inverse of x is: a) x+1 b) x2 +1 c) x d) x? 2- The value that replacesx is…

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Q: Let G be a direct product of H and K then Z(G) is a direct product of Z(H) and Z(K).

A: By the set inequality i have solved the problem

Q: Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (х, у)…

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Q: Use Gauss-Jordan method to calculate the inverse of A A-l = |1 0 -1 A =| 0 1 1 1 1 1

A: By Gauss Jordan Elimination method the inverse of matrix is given as follows :

Q: If El and E2 are sigmas algebras over X, is ElU E2 are sigma algebra over X? Prove your answer.

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Q: 2) Show that additive inverse of any z = (x, y) E Cis unique and it equals-z = (-x, -y). Show that…

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Q: If T is self-adjoint, then S*TS is self adjoint for all S; if S is invertible and S*TS is…

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Q: 1. By mathematical induction, prove that if A,, A2, .., An are invertible, then the product A,A2 ...…

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Q: Evaluate det(A) by a cofactor expansion along a row or column of your choice. 6 0 0 1 01 2 2 2 -1 0…

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Q: Exercise 5.4.18. Give a similar proof that 1 is the multiplicative identity for Zn when n > 1. What…

A: To show 1 is the multiplicative identity of  ℤn for n>1 and 0 is the multiplicative identity of…

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: Evaluate det(A) by a cofactor expansion along a row or column of your choice. 3 3 5 2 0 -2 (a) A= 4…

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Q: In the ring Z[x] 1- The inverse of x is: a) x+1 b) x+1 c) x d) x 2- The value that replacesx is a)…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Prove that if gcd (a,b)=1>then gcd (ab,ab)=l

A: I have proved using the fact that gcd can be expressed as integers linear combination q

Q: Question 5. (a) Prove that every element in Q/Z has finite order. (b) Prove that every non-identity…

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Q: In the ring Z[x] 1- The inverse of x² is a) x+1 b) x2+1 c) x d) x2 2- The value that replaces x is…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let * be the operation on the cartersian product of real numbers R *R such that (a,x) * (b,y) =…

A: Given that, let * be the operation on the cartesian product of real numbers R*R  such that (a, x) *…

Q: Prove that if Σ1 and Σ2 are sigma algebras over X, then Σ1 ∩ Σ2 is a sigma algebra over X.

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Q: Determine which elements of Z8 have inverses under the operation ×8. Then do the same for Z7 under…

A: For an element a a multiplicative inverse is an element b such that ab=1. The set Zn is the set of…

Q: Question 9. (a) Prove that every element of Q/Z has finite order.

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering the…

Q: Compute only one the inverse z-transform of 02 a) X(z) = 5)X(z) = Q3 ROGIZ| >3- ROC Z > 3 %3D %3D…

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Q: (a) Prove that if ), and > are segma algebras over X, then >n) is a sigma algebra over X. (b) Is a…

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Q: (1) Since (1)(1) - (0)() = 1- 0 = | Hence IE G La ea be any element a b b. Now identity so be the…

A: Follow step 2nd .

Q: show that the set Z ={0,+-1, +-2,...} with the operation of multiplication (a,b) map to ab is not a…

A: As per the bartleby guideline I can solve only 1st question or 3 sub questions of 1st question if…

Q: 2) Show that additive inverse of any z = (x, y) E C is unique and it equals-z = (-x, -y). Show that…

A: Let additive inverse of Z = (x14) E C is (a,b) Then, Z + (a,b) = (0,0) As (0,0) is identity in…

Q: Let G be a direct product of H and k product of 2 (H) 'and z (k) then Z(G) is a direct

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Q: Prove Theorem 2.30 Consider the rule for multiplication in Z„ given by [a][b] = [ab] a.)…

A: PART A- Given: The rule for multiplication in ℤn given by ab=ab We want to show that multiplication…

Q: Compute the Killing form for the Heisenberg algebra and find its radical.

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Q: 6. Prove that if gcd(a, b) 1, then gcd(a +b, ab) = 1. %3D %3D

A: Given that gcd(a,b)=1 If possible , let us assume that  gcd(a+b,ab)=d,d≠1 Then the number d must…

Q: Prove that every element of Sn (n > 1) can be written as a productof elements of the form (1k).

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Q: Show that for any k-algebras A1, A2,..., A,, we have (A1 x A2 x ...x A, )P S A" x A" x as algebras.

A: Let, A1,+,*,A2,+,*,...,Ar,+,* be the K-algebras, where + is the addition from 1 to r o which A1 is…

Q: Prove that Sn is not abelian for all n > 3.

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Q: 4 Let [a] e Z. Use Theorem 212 to prove if a and n are relativelyprime then [a] has a multiplicative…

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Q: 2. (a) Prove that if E, and , then >n, is a sigma algebra over X. (b) Is E,UE, are segma algebras…

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Q: Prove that is G1 is isomorphic to G2 and H1 is isomorphic to H2, then G1 x H1 is isomorphic to G2 x…

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Q: > a sigma algebra over X ?

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: QUESTION 3 Let o be a homomorphism from Z36 onto Z4. a) Determine the kernel of o. b) Find all the…

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Q: The expansion of esin(x) is Select one: O a. x² x4 1+x+ ... 8 Ob. x² 1+x+- 2 - t... 8 c. x5 1+x+ 6.…

A: To find the expansion of

Q: Let a ∈ R[[x]] with a0 not equal to 0. Prove that a has a multiplicative inverse in R[[x]]. You may…

A: We have given that , a∈ℝx a = a0 + a1x + a2x2 + ......  where each ai ∈ ℝ. Since , 1 ∈ ℝx  is an…

Q: Q2. (a) Prove that if), and >, are segma algebras over X, n, is a sigma algebra over X. then a sigma…

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Q: (b) Show that the function : Z12 → Z12 defined by (a) = a + a is a homomorphism. (c) Find the kernel…

A: We know that a mapping ϕ:G, ∘→G', ∆ is a homomorphism if for all a, b∈G, we have, the property…

Q: b) Let 0: Zg – Z4 given by Ø(x) = 2x Show Ø is homomorphism %3D Determine the kernel of Ø Determine…

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Q: Let f : (-7, "] → R f (x) = 7 if - R < x < 0 _and S (x) = 7 if 0<x < m. Determine bị in the Fourier…

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Q: 5. Let M22 be the ring of all 2 x 2 matrices with respect to matrix addition and multiplication, and…

A: Solution by using subring criteria : Equivalently: The criterion for a subring A non-empty subset S…

Q: (a) Prove that if , and 2 are segma algebras over X, then nL, is a sigma algebra over X. (b) Is ),…

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Q: 4. Let w = e, and let E = Q(w). (a) Verify that [E : Q] = 10. (b) Determine |Gal(E/Q)|, the number…

A: Since you have asked multiple questions, we can solve first question for you. If you want other…

Q: 9. a) find two integers x,y such that 30x+101y=1. (hint: gcd(30,101)=1).     b) find the inverse of…

A: Given: We have given numbers 30 and 101 We want to find integers x and y such that 30x+101y=1

Q: 1 1 -1] C= 2 0 2 , rank(C)=2 101

A: Det(C) = 1(0-0) - 1(2-2) + (-1)(0)            =0