2. Prove that f: (R × R, +) → (R,+) given by f(x, y) = x + y is a homomorphism. Is g: (R × R, +) → (R, +)given by g(x, y) = x.y a homomorphism? Explain your answer.(Hint: Check whether f((a, b) + (c,d)) = f(a,b) + f(c,d).)

Answered: Image /qna-images/answer/6f1846c6-810b-40f8-96b7-0566f2d3e431.jpg

Answered: 2. Prove that ƒ : (R × R, +) → (R, +)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Similar questions

Define an equivalence relation on R² by: ¤1, Y1) ~ (x2, Y2) E if and only if (x1, Y1) and (x2, Y2) have the same distance from the origin. (a) Interpret the equivalence classes geometrically and describe the quotient R?/ ~. (b) Is the function f: (R²/ ~) → R, [(x1,y1)] → xỉ + yỉ well defined?
f: X→ Y and g: Y→V be bijective functions. Show the following: ) gof: X → V is injective. ) gof: X→ V is surjective. ) (gof)-¹ = f¹og-¹ v injective if x1 #xn → v(x1) = v(xn)
1 The coordinate of the invariant points when f(x) = -x + 2 is transformed to y = √√(x) are: (a,b) and (c,d). State the values of a,b,c and d in the answer box (no spaces, no commas: abcd)
c) Let the set A = {-1,0, 1} and consider the following function: f3 A × A → A × A given by f(x,y) = (y7,x). Decide whether this function is injective, surjective, a bijection. f3 is an injection: Not Answered • ƒ3 is a surjection: Not Answered f3 is a bijection: Not Answered d) Let the set A = {1, 2, 3,4} and let P(A) denote the power-set of A. Consider the following function: f₁ : P(A) → P(A) given by ƒ(B) = {1, 2, 3}\B. Decide whether this function is injective, surjective, a bijection. f4 is an injection: Not Answered f4 is a surjection: Not Answered • f is a bijection: Not Answered
15. Let f: A → B and g: C → D. Define fx g = {(a, c), (b, d): (a, b) = f and (c,d) = g}. (a) Prove that f × g is a mapping from A × C to B x D. (b) For (a, c) = A x C, write (f× g)(a, c) in terms of f(a) and g(c).
Consider the relation {(x, y)|x, y E R, (x – 2)² +y² = 1}. (a) Does the above relation represent a function? Graph the relation and justify your answer. (b) Which of the following symmetries are respected by the above relation? symmetry about x-axis, symmetry about y-axis, symmetry about the origin. Justify your answer.
g) Prove that f : R → R where f(x) = x² – x is not injective. h) Disprove the following claim by counter example. "For all natural numbers x, x² is odd". Show your work

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2. Prove that ƒ : (R × R, +) → (R, +) given by f(x, y) = x + y is a homomorphism. Is g : (R × R, +) → (R, +) given by g(x, y) = x - y a homomorphism? Explain your answer. (Hint: Check whether f((a, b) + (c,d)) = f(a, b) + f(c,d).)