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).)

Q: Let S : RR be defined by S(x1, x2, 3,...) = (T2, T3, T4,...) (a) Using the definition of linear…

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Q: 5. Determine whether the function f : R+ → R defined by the rule f (x) = positive real numbers x in…

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Q: f: X→ Y and g: Y→V be bijective functions. Show the following: ) gof: X→V is injective. ) gof: X→ V…

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Q: (5.9) Example: Prove f: R² x R? →, define by f((x1,y1), (x2,Y2)) = x1X2 – YıYz is symmetric bilinear…

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Q: We denote the set of polynomials with real coefficients as R[X] := {ao + a₁X + a₂X² + ... + a₂X": n…

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Q: III. For the given function f and g, find the composition of the following: a) f •g b) gof c) fof d)…

A: Since, you have asked multiple sub parts questions, we will solve the first two questions. If you…

Q: f: X→Y and g: Y→V be bijective functions. Show the following: ) gof: X→ V is injective. ) gof: X → V…

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Q: Let C = {0, 1, 2, 3} and f: C → C, g: C → C with ƒ = {(0, 2), (1, 2), (2, 3), (3, 0)}, {(0, 3), (1,…

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Q: Q4. Let X = {m, n, p, q}, T = {4, X, {p}, {m, n}, {m, n, p}} and Y = {a, b, c, d}, F = {ø, X, {d},…

A: There is a slight error in the question. X cannot be an element of F. Instead of X, it will be Y.…

Q: Let T = {0, 1, 2, 3} and the relation R is defined on T as follows: R = {(0, 1), (1, 2), (2, 2)}.…

A: Given  T = {0,1,2,3} and the relation R is defined on T as follows R= {(0,1),(1,2),(2,2) }

Q: Let f:R → R such that f(x) = x² – 2x + 1. Verify whether f is bijective ? explain your answer.

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Q: b) A function g:Z→Z is defined by the rule g(m, n) = m + n Determine whether it is onto or not...

A: A function f:A→B is said to be onto or surjective if for all element b∈B there is an element a∈A…

Q: Let f: Z/4Z → Z/2Zx Z/2Z be defined by f([0])= ([0], [0]), ƒ([1]) = ([1],[1]), f([2]) = ([0], [1]),…

A: We see,         [2] = [1] + [1]    but, f([2]) = ([0], [1])          f([1]) + f([1]) = ([1], [1]) +…

Q: a-c

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Q: Define a bijective map g : x2/3--->R.

A: Sol

Q: f: R² → R2 with f(x, y) = (2x, x+y) Which statement is correct? f is NOT a linear map and it is not…

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Q: Please find a relation R that makes the statement true and a relation R(x, y) that makes the…

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Q: Let f, g,h: A→ A be any bijective functions. Then (f ogoh)-

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Q: Suppose a transformation has a mapping as follows:(x,y)→(1/2 x + 3, -2y + 4). If this mapping were…

A: Given: A function y=x and a transformation mapping x, y→12x+3, -2y+4.

Q: is bijection. Do the same thing with Z,.

A:

Q: 1. L: V3 V4 defined by L(a, b, c) = (a - b,b+c-d, a+b-2d, b-c-d) 2. L: V4 → V3 defined by L(a, b, c,…

A: According to Bartleby guidelines we can only solve 1 question or three subparts. So put other one…

Q: 11) Let f: IR4 → IR³ be defined by : IV f(x₁y₁z₁t) = (2x+y-t, x+8y-92- 2t, x-2y + 3z) (a) show that…

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Q: Show whether or not f is bijective. Then, give the range of f. a. Let f: ZN be defined as f(x)=…

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: c) Let the set A = {-1,0,1} and consider the following function: f3 A × A → A × A given by f(x, y) =…

A: A function f is said to be injective or one-one if the function value corresponds to only one point…

Q: A relation R on a set A is called irreflexive iff (Vx E A)((x, x) ¢ R). It is called asymmetric iff…

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Q: Exercise 3.13. Let f : AB be a one-to-one correspondence. By Exercise 3.12, f-1: B → A is also a…

A: see the attached image below

Q: Which of the following relations are functions of y in terms of x? 1) 2) 3) 4) 5- 5+ 5+ 4- 3- 2- 2-…

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Q: Let X be the set {5, –6, 2}. For the first three parts of this problem you are asked to define a…

A: Consider the set 5,-6,2. The function is f:X→X and the relation is u~w⇔w=fu.   a) The relation is…

Q: Consider the function g : ZxZ → Z × Z defined by g(x, y) = (4xy, 9x + 5y). (a) Is g injective? (b)…

A: (a) we see (5, 0) and (0, 9) are different elements in ZxZ.      But, g(5, 0) = (4.5.0, 9.5+5.0) =…

Q: P1=(x1,y1,z1) and P2=(x2,y2,z2)  prove that distance of P1 and P2 ; |P1P2|=((x2-x1)2 +(y2-y1)2…

A: Solution: The objective is to prove that the distance of P1 and P2   |P1P2|=((x2-x1)2 +(y2-y1)2…

Q: Prove or disprove whether the function f (r) = a +2 is a bijection from R R %3D

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Q: 4) Let ƒz = dƒ/əz and ƒz = 0ƒ/əz. Recall ƒ = u + iv, the notation f is used for the complex…

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Q: 1 The mapping (x, y)(x+h, ay + k) can transform 1 y= f(x) into y +2 +7f((x+3)). What would be the…

A: Kindly do rate this  helpful.Thank You