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

A:

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

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

Q: 35 Let I(m, n) be a function that satisfies the relation I(m, n)m + I(n, m)n gcd(m, n), when m and n…

A: It is given that Let I(m, n) be a function that satisfies the relation I(m, n)m+ I(n, m)n = gcd(m,…

Q: Let A = R - {1} and define f: A→A by f(x) = (a) Find fo f(x). (b) Find fofo f(x). (c) Find f-¹(x).…

A:

Q: A consumer has set a budget of $200 for the consumption of good X and Y. The price of Good X is $30,…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

A: According to the given information,

Q: Let g : B → N × N where B C Z given by g(m) = (|m| + 2, |m| + 3). Choose all subsets B such that g…

A:

Q: 22

A:

Q: 1. Let Q(v2) = {a+b/2| a, b E Q}. Prove that Q(/2) = Q[x]/(x² – 2).

A: We have to show that isomorphism means there is function ϕ such that we have to show that it is an…

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

A: The first part of the question is asking us to prove that the sum of the kth powers of the first n…

Q: 23. Determine whether each of these functions is a bijection from R to R. a) f(x) = 2x + 1 b) f(x) =…

A:

Q: 2. Let (r) be the function defined by V(x) = 1. Let 1+x. In(x) Vovo...(x) n-copies Note that…

A:

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]…

A: The objective of the question is two-fold. First, we need to find a unit in the ring Z[sqrt(2)] that…

Q: 3. Determine whether each of the following functions is a bijection and explain why. (a) f: R R,…

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

Q: (2) Let f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

A:

Q: 6. Let f(x) = x55x+1. Prove that f has at least 3 zeros in [-2, 2]; that is, there exist distinct…

A:

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

A: The first part of the question is asking us to prove that the sum of the kth powers of the first n…

Q: Show that the function f: N² → Z given by f(m, n) = m²-n² is neither injective: suriective. nor

A:

Q: 1. Let A={1,3,5,7}. Let f:A→Z be the function f={(1,2),(3,4),(5,6),(7,8)}. Let g:Z→Z be defined by…

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…