Consider the set Z, × Z, = {(x, y) : x € Z, and y € Z3}. Define the operation/sum O in Z2 × Z3 such that for any two elements (a, b) and (c, d) in Z, x Z3, we have (a, b) (c, d) = (a +2 c, b +3 d) where +2 is the addition modulo 2 in Z2, and +2 is the addition modulo 3 in Z3. 1. a. List the element of Z, x Z3. b. Determine the identity element in Z, × Z3. c. Determine the inverse element of each element in Z, x Z3. d. Is Z, x Zz a group under the given operation? Why? Is it Abelian? e. List a proper subgroup of Z, × Z3. Write "none" if there are no proper subgroups.

Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

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Q: 5. Verify if the following are difference sets and state v, b, k, r, A for the ones that are: a)…

A: 5(a) Given that, (0,8,15,17,20,21)mod31 The cardinality of the set is 6 with mod31. Here V is the…

Q: Let a, m € Z+ and (a,m) = (a - 1, m) = 1. Then prove that 1+a+...+ a(m)-1 = 0(modm)

A: For given,       (a,m) = (a -1, m) = 1 We have to show,      1 + a + a 2 + ........ + a φ (m)-1 ≡ 0…

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Q: 2. Let a be a positive integer. Explain why the linear congruence ax = b (mod a + 1) has a solution…

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Q: 18. In each part, let T₁: R3-R² be multiplication by A, and let u,= (0, 1, 1) and u₂=(2,-1, 1) and…

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Q: The permutations p = (3 5 4), q= (1 5 2 3) and r = (1 5 2) are elements of S5. pal, m. (a) Find each…

A: (a) To find each of the given permutations in cycle form, we will apply the permutations…

Q: a) Find integers a and b such that a + b =a-b (mod 5).

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Q: Please answer letter D and E onlyyy.

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Q: The linear congruence ax = b(mod m) has a unique solutionif and only if: gcd (a,m) =1 gcd(a,m) =1 o…

A: First option is correct. gcd(a, m)=1.

Q: Prove that the operation * defined on M2x2(R) by taking A * B = A. B V A,B E M2x2(R). %3D is an…

A: Given that A*B=A.B ∀ A, B∈M2×2(R)

Q: Let there be set of a, b, c. Prove A×(B∩C) = (A∩B)×(B∩C)

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Q: Q2a) For all integers a,b,c, prove or disprove each of the following: i) If a | c and b | c, then ab…

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Q: Prove that (A ∪ B) − A = B − (A ∩ B) using: (a) the choose-an-element method (b) the algebra of set

A: We have to prove the set using two methods.

Q: Let a = (1 73)(5 49 2) and B = (2 3)(7 4)(5 18) be elements of S10. Then, a B is O (12974)(3 58) O…

A: Let α=1735492 and β=2374518 be elements of S10, then α2β-1 is

Q: The coefficient of x³ in (1 + x² + x² + x6)4 is determined by the 4-tuples o integers (a1, a2, a3,…

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Q: Prove that ∀a, b, k ∈ Z that ((4|k) ∧ (k = ab)) ⇒ (2|a ∨ 2|b)

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Q: Let the sets A, each bo countable. (a) Show that the countable union U₁A; is countable (6) Show that…

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Q: Suppose that 2 = {1,2,3,4} and C = {{1}, {2}, {3,4}}. Determine o(C), the smallest o-algebra on 2…

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Q: Let S be a set defined recursively as follows Basis Step: (1,1) e S. Inductive Step: (a, b) ES (a +…

A: Let S be a set defined recursively as follows, Basis step: 1,1∈SInductive…

Q: A universal​ set, with​ n(U) = 30​, is partitioned into three​ subsets: A,​ B, and C.     If​ n(B) =…

A: we are given n(U) =30​, n(B) =3​⋅n(A), and​ n(C) =2​⋅n(B) also universal set is partitioned into…

Q: The linear congruence ax = b(mod m) has a unique solutionif and onlyif: @cd(a,b) = 1 cd(a,m) =1…

A: Second option is correct.

Q: A cosets of of 212 are اخترأحد الخیارات + 2, +1,.1 O + 1,.2 O + 4, + 1, .3 + 4, + 2, + 1 , .4 O

A: Cosets Definition: Let G be a group and let H is a nonempty subset of G. A left coset of H in G is a…

Q: Suppose that each of A, B, and C is a set and each of BnC, A–(BnC), A-B, A-C, and (A – B) n (A – C)…

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Q: If gcd(7, a) = 1, prove that either a +1 or a – 1 is divisible by 7. 20

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Q: Determine the orders of the elements of D∞

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Q: Are the polynomials x³+2x +2 and x4 + 3x -2 equal to mod(x² + 2x + 2) in Q[x]?

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Q: In the following list, select all the correct statements, if (4,) is a poset and for any elements…

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Q: 1. List a pair of zero divisors of (M₂(Z), +, ×):

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