Let A := 1+√5 2 and B := 1−√5 2 . These are both roots of...

Let A := 1+√5 2 and B := 1−√5 2 . These are both roots of the equation x2 = x + 1. Show that fn = An −Bn √5 , where fn is the nth ﬁbonacci number.

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Colby Messinger answered 1 month ago

6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f...

6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f = {(1, 5),(2, 5),(3, 6),(x, y)} where x ∈ A and y ∈ B are to be determined by you. (a) In how many ways can you pick x ∈ A and y ∈ B such that f is not a function? (b) In how many ways can you pick x ∈ A and y ∈ B such that f : A → B...

(2) Let ωn := e2πi/n for n = 2,3,.... (a) Show that the n’th roots of...

(2) Let ωn := e2πi/n for n = 2,3,.... (a) Show that the n’th roots of unity (i.e. the solutions to zn = 1) are ωnk fork=0,1,...,n−1. (b) Show that these sum to zero, i.e. 1+ω +ω2 +···+ωn−1 =0.nnn (c) Let z◦ = r◦eiθ◦ be a given non-zero complex number. Show that the n’th roots of z◦ are c◦ωnk fork=0,1,...,n−1 where c◦ := √n r◦eiθ◦/n.

solve for matrix B Let I be Identity matrix (I-2B)-1= 1 -3 3 -2 2 -5...

solve for matrix B Let I be Identity matrix (I-2B)-1= 1 -3 3 -2 2 -5 3 -8 9

(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2....

(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2. Prove that if 2p−1 is prime, then p must also be prime. (Abstract Algebra)

Let X = {1, 2, 3, 4, 5, 6} and let ∼ be given by {(1,...

Let X = {1, 2, 3, 4, 5, 6} and let ∼ be given by {(1, 1),(2, 2),(3, 3),(4, 4),(5, 5),(6, 6),(1, 3),(1, 5),(2, 4),(3, 1),(3, 5), (4, 2),(5, 1),(5, 3)}. Is ∼ an equivalence relation? If yes, write down X/ ∼ .

a.) Prove the following: Lemma. Let a and b be integers. If both a and b...

a.) Prove the following: Lemma. Let a and b be integers. If both a and b have the form 4k+1 (where k is an integer), then ab also has the form 4k+1. b.)The lemma from part a generalizes two products of integers of the form 4k+1. State and prove the generalized lemma. c.) Prove that any natural number of the form 4k+3 has a prime factor of the form 4k+3.

Let Z[ √ 2] = {a + b √ 2 | a, b ∈ Z}. (a)...

Let Z[ √ 2] = {a + b √ 2 | a, b ∈ Z}. (a) Prove that Z[ √ 2] is a subring of R. (b) Find a unit in Z[ √ 2] that is different than 1 or −1.

Theorem 3.4. Let a and b be integers, not both zero, and suppose that b =...

Theorem 3.4. Let a and b be integers, not both zero, and suppose that b = aq + r for some integers q and r. Then gcd(b, a) = gcd(a, r). a) Suppose that for some integer k > d, k | a and k | r. Show that k | b also. Deduce that k is a common divisor of b and a. b) Explain how part (a) contradicts the assumption that d = gcd(b, a).

Let a and b be integers which are not both zero. (a) If c is an...

Let a and b be integers which are not both zero. (a) If c is an integer such that there exist integers x and y with ax+by = c, prove that gcd(a, b) | c. (b) If there exist integers x and y such that ax + by = 1, explain why gcd(a, b) = 1. (c) Let d = gcd(a,b), and write a = da′ and b = db′ for some a′,b′ ∈ Z. Prove that gcd(a′,b′) = 1.

4) Let ? = {2, 3, 5, 7}, ? = {3, 5, 7}, ? = {1,...

4) Let ? = {2, 3, 5, 7}, ? = {3, 5, 7}, ? = {1, 7}. Answer the following questions, giving reasons for your answers. a) Is ? ⊆ ?? b)Is ? ⊆ ?? c) Is ? ⊂ ?? d) Is ? ⊆ ?? e) Is ? ⊆ ?? 5) Let ? = {1, 3, 4} and ? = {2, 3, 6}. Use set-roster notation to write each of the following sets, and indicate the number of elements in...

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