5. Let d € Z - {0}, assume that √d & Q, and let σa : Z[√d] → Z[√d] be the conjugation function. (a) Prove that σd(21 +22) = Od(²1) + Od(22) for all 2₁, 22 € Z[√d]. (b) Prove that (²1²2) = 0d(²₁)0d(32) for all 21, 22 € Z[√d]. The formulas established in (a) and (b), along with the fact that σa(1) = 1, amount to the assertion that oa is a ring homomorphism (for those with linear algebra experience: linear transformations are to vector spaces as ring homomorphisms are to rings).

Q: Prove that there does not exist any function f(x, y) such that

A: To show that there does not exist any function that satisfies the given conditions

Q: 7. Prove that the function f:Z → Z defined by f (x) = x + 8 is one-to-one and onto.

A:

Q: (b) Let x € R. Prove that if x² – 2x − 3 ≥ 0, then x² + 1 > 0. -

A:

Q: Prove that f(x) = 5x2 + 3x – 26 is θ(x2)

A: Given: f(x) = 5x2 + 3x – 26 is θ(x2)

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A:

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A: According to the given information,

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A:

Q: 22

A:

Q: a) Let f: Z Z be such that f(x) 2*x. Is f invertible, and if so, what is its inverse?

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

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A:

Q: 1. Prove that Vk Є N, 1k+2k + ·+nk € © (nk+1). 2. Suppose that the functions f₁, f2, 91, 92 : N →…

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Q: Q2: Let T₁, T₂, be the following bounded linear operators ¹ → T₁(x₁, x₂. x3,) = (x₁, X₁, X₁, X₁,…

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Q: 1. Determine whether each of these functions is a bijection from R to R a) f (x) = 5x + 10 B) f (x)…

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Q: 18. Let A = {1,2, 3}, B = f: A → B, and g : B → C, such that {W, Y, Z}, C = {7, V2, V5}. Find…

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Q: 3. (a) Find a unit in Z[√2] which is not equal to 1 or -1. Hint: Use 1 (b). (b) (*) Show that Z[√2]…

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A: Note:As our company  guidelines we are supposed to answer only first 3 sub-parts. kindly re-post…

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A:

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Q: 1. Prove that Vk Є N, 1k+2k + ·+nk € © (nk+1). 2. Suppose that the functions f₁, f2, 91, 92 : N →…

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Q: 21. Let f(n) = [/n] – [/n – 1]. Prove that f is multiplicative but not completely multiplicative.

A: See the attachment

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A: Given, Set A=1,3,5,7 and a two functions f and g as follows, f: A → Z by f=1,2,3,4,5,6,7,8 g: Z →Z…