Fix a E Z. Show 9a : Z → Z defined by Qa(n) = an is a homomorphism.

Q: Let fg,:I→R2 begivenby f(x) = (0, x), g(x) = (1, x). a. Specify a homotopy h:I × I→R2 from f to g.…

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Q: 3. Suppose that ged(m, n) = 1. Define f : Zn Z x Z, by f(r]mn) = ([T]m; [7]n). %3D (a) Prove that f…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Prove that H has zero divisors.

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Q: ~ y iff f(x) = f (y). 5. Let f : X → Y be a surjective function. Define a relation on A by setting x…

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Q: 6. Give an example of a relation R on A = {-1, –3, –5} such that: (a) R is both symmetric and…

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Q: 17. If E is an extension of F and f (x) e F[x] and if is an automorphism of E leaving every element…

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Q: Prove that the mapping from U(16) to itself given by x → x3 is anautomorphism.

A: To show that the mapping from U16 to itself given by x→x3 is an automorphism, we need to show that…

Q: For all v € V, let : V' → F be defined by (p) = y(v) for y € V'. (a) Show that is linear, i.e. € V",…

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Q: Assume that

A: Given, X=a, b and  *: X2→X is such that                        x * y * z=z * x * yfor all x, y, z∈X.…

Q: 1. Consider the matrices A₁ 4 = ( 1 ) - 1 3 0 A₂ 42 = (3-2) ₁ 4₁ = ( 2² -2²), , A3 0 -2 These…

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Q: If h:(R,+) → (R*,.) is a mapping defined by h(x)=10*, VXER, then (R,+) = (R*,.).

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Q: Let T : V → U be a homomorphism, then Ker T = { 0 } iff T is one-one

A: A linear transformation is said to be injective or one to one if provided that for all v1 and v2 in…

Q: Show that the set A = {fm,b : R → R | m # 0 and fm,b(æ) group under composition of functions. (The…

A: To Show- Show that the set A = fm,b : R → R | m ≠ 0 and fm,bx = mx + b, m, b ∈ R forms a group under…

Q: Exercise 3. Show that f : Z → N so that f (z) = z² + 1 for all z E Z is neither one-to-one %3D nor…

A: One-One Function: A function is said to be One-One if for every element in the domain there exists…

Q: a T:R' → R2 [c] is given by T 3 (а — ) + (с — b)ӕ + (с — d) 2?. Which d statement about T is…

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Q: Let V = V(r - y) C A. Consider the morphism o: A' → V given by o(t) = (t,t). (a) Show that o is a…

A: Given: V=Vx3-y2⊂A2 ϕ:A1→V by ϕt=t2,t3 To do: Show that ϕ is a bijection but not an isomorphism.…

Q: Let p: Z → Z30 be the map p(x) = 21x. 1. Show that p is a ring homomorphism. 2. Find Ker(y). 3. Find…

A: Definition of Homomorphism:        It is a structure preserving map between two algtebraic…

Q: Let p: GL(2, R) → R' be a homomorphism map defined by p(A) = det(A), then %3D a) Ker o = GL(2, R) b)…

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Q: a T: Raz]M2x2(R) is given by T(a+ bx + ca?) = (, %3D b b+c-a, Which statement about Tis correct? O…

A: Given transformation is linear transformation, T:ℝ2x→M2×2ℝ s.t Ta+bx+cx2=acbb+c-a. Let,…

Q: Construct a nontrivial homomorphism $: Zg → Zı5 >

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Q: 8 w y - 1 A UIE TNUerse of 7. We then have:. B has an inverse if and only if it is bijective,…

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Q: Let D= {1, 2, 3, 4). Give an example of a relation W on D such that W is anti- symmetric but not…

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Q: 6. Classify the given mapping y: A→B by checking 5 properties. Each property must be explained. y :…

A: Sol:- The given mapping y: A → B, where A = (−∞, +∞) and B = [−5, +∞), can be classified as follows:…

Q: Suppose (X,≺) is a well-ordering and f : X → X is an isomorphism. Then f is the identity function on…

A: Well ordering

Q: Let : [0, 1] → [0.c], te- and let w be a 1-form on R. What and foe, and why? is the relation between…

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Q: In each question, calculate dg, and decide whether dg, is an isomorphism. If dg, is an isomorphism,…

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Q: 6. Define the function E : Z → Z by E(n) 2n. Prove that E(n) is one-to-one, but not onto.

A: We will use the definition of one to one functions and onto functions to prove this.

Q: Let F be a field and let a: F³ → F[X] be the linear transformation defined by રી :] b с A. Find the…

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Q: Define T: C² C² by T((w, z)) = (iw, z-w). Show that T is an isomorphism, s, show that T is linear,…

A: We need to show that T is linear,oneone,onto

Q: Suppose  φ: ℤ x ℤ→ℤ defined by φ(m, n) = m - n. a. Show that φ is a homomorphism b. Compute Ker φ…

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Q: Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following…

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Q: Construct an isomorphism from R into RT. Hint: what is the inverse function of In (x)? Prove that…

A: Homomorphism:- Let G1,+1 and G2,×2 be any group.A function f:G1→G2 is homomorphism if fa +1 b=fa ×2…

Q: Exercise 3. Prove that a relation R on A is antisymmetric if and only if RNR-' C {(a, a) | a E A

A: We know that , A relation R on set A is antisymmetric  if either x , y∉ R  or (y , x) ∉ R whenever…