6. If : G H and 8: H K are both homomorphism, so is 0. • : G → K.

Q: rphisms : Z2

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Q: Explain why the correspondence x → 3x from Z12 to Z10 is not ahomomorphism.

A: We have to explain that: Why the correspondence x→3x from Z12 to Z10 is not a homomorphism.

Q: 5. Let a E G be fixed. Define y: G → G by p(x) = axa-1 for all x E G. Prove that p is a…

A: To find - Let a∈G be fixed. Define ψ:G→G by ψx=axa-1 for all x∈G. Prove that ψ is a homomorphism.

Q: Determine all homomorphisms from ZO Z2 to S3.

A: Z×Z2→S3Z has subgroups like  mZ   where m∈ZZ2 has subgroups 0,Z2now apply fundamental theorem of…

Q: You are given two graphs G and H with V(G) = {a, b, c, d} and V(H) = {1, 2, 3}. 1. How many…

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Q: Disprove that for any relations R and S, ran(S • R) = ran(S).

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Q: Consider the relation being described by the following arrow diagram. Which of the these statements…

A: (i) It is given that ;      S = {6, 7, 8}  and  T = {2, 4, 6} And the relation between S and T is :…

Q: 1. Determine whether R = {(x, y) | x + y = 0} on the set of all real numbers is reflexive,…

A: Since you have posted a multiple question according to guildlines I will solve first(Q1) question…

Q: Determine all homomorphisms from D5 onto Z₂ Z₂. Determine all homomorphisms from D5 to Z₂ + Z₂.

A: We have to determine all homomorphism from  onto .We also have to find homomorphism from  to

Q: Let A = {2" : m e Z}. A relation - is defined on the set Q+ of positive rational nu a a ~ b E A is…

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Q: 5*. Suppose that 0: G -+ H is a homomorphism. Prove that (a) (a")= (x)" for all positive integers n.…

A: To prove the properties of group homomorphisms

Q: Let f: X → Y and g : Y → Z be arbitrary maps of sets (a) Show that if f and g are injective then so…

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Q: (2.3) Prove that R is an equivalence relation.

A: Prove R is an equivalnce relation.

Q: 4. Define a relation R on N by aRb if a and b have no common factors except 1. (a) Give an example…

A: We have define  a relation R on natural numbers (N) by aRb if a and b have no common  factors except…

Q: Let G=U(16),H={1,15},andK={1,9} . Are H and K isomorphic?Are G/H and G/K isomorphic?

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Q: Determine the orders of the elements of D∞

A: It is known that, D∞=a,b|a2=b2=e also it can be observed that, D∞=abka,abk,bak,bakb Let x∈D∞. If…

Q: E. Let o : R+ → R* under multiplication be given by o(x) = |x|. 1. Show that is a homomorphism. 2.…

A: ϕ: R+→R+ϕ(x)=x

Q: Build an isomorphism between M2  (R) and P3  (R)

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Q: (a) Let eg be the identity element of G, let e be the identity element of H, and let g E G. Prove…

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Q: 7- cosy dx+ (x siny-cos'y)dy=0

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Q: Which of the following is isomorphism? f:(Z, +) → (Z, +) where f(x) = 2x. this option f: (R, +) →…

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Q: 1- show that [Am, A¹] = 0 for all m and n discuss the physical meanings of this relation. 2-…

A: As per the question we are given the following commutation operations : [Am , An] [∂/∂x , x] And…

Q: . Define a relation R on Z as follows: a Rb if and only if 5 | (a²- 6²). Find four integers which…

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Q: 9. Determine whether f is injective, surjective or bijective. a. Suppose f: N→ N has the rule f(n) =…

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Q: 7. Let A and B be sets, and let f: A → B and g: B → A be functions. Suppose that: • go f(a) = a for…

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Q: 3. whether Gand G'are isomorphic. determine of G G'

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Q: 2. Let A := {1,2,3,4} =: B and T:= {(a, b) E A x B | ª+b+¹ € Z}. (a) Find A × B. (b) Find T. (c)…

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Q: 4   1. Define the relation | on N\{0, 1} by a | b precisely when a divides evenly into b.   (b)…

A: Given relation I on N-{0,1} fora,b∈N-{0,1} a∣b⇒b=La for l∈ℕ ∵a∣a  ∀a∈N-{0,1} ∴ I is Reflexive.

Q: e) Define W:Z → Z and V:Z → Z by the rules W(a) = a² and V (a) = a mod 5 for all integers %3D a.…

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Q: Define a relation R on N by a R b if and only if a and b have a common rime factor (for a, b E N).…

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Q: For every element x in A, there is an element in y is B such that (x,y)∈F. QUESTION: How does this…

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