cyclic

Q: Prove that Eo 2i = 2n+1 – 1 - i3D0

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Q: Let G = {x ∈ R : x != −1} . Define △ on G by x△y = x + y + xy.   (a) Show that △ gives a binary…

A: (a) G={x∈ℝ: x≠-1}.THe operation is given by x∆y=x+y+xyHere, since ℝ is closed under addition and…

Q: State and prove Euclid’s Lemma.(you may assume without proof that for integers x and y there exist…

A: If x and y are integer not both zero then there exist integer k and l such that kx+ly=gcdx,y

Q: If El and E2 are sigmas algebras over X, is ElU E2 are sigma algebra over X? Prove your answer.

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Q: Prove B = {1/n : n in Natural numbers} is a union of {0} and that it is compact.

A: A set B is said to be compact if every open cover has a finite subcover of B.

Q: Determine all s ∈ R such that I4s= (I4)s

A: Given I4s=(I4)s We know I4=1 I4s=(1)s

Q: Compute for (1+u+ u²)(1+u).

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Q: Prove that Zn has an even number of generators if n > 2. What doesthis tell you about Φ(n)?

A: To Do: We have to prove that Zn has an even number of generators if n>2. Also, discuss what does…

Q: Which of Zs, Z2o are cyclic? 18, '20

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Q: left cosets

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Q: This exercise provides an alternative to Theorem 8.2 for factoring an r-cycle into transpositions.…

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Q: Several friends chartered a boat for a day's fishing. They caught a total of 67 fish. The table…

A: We will determine the empirical probability that the next fish caught is a kingfish. From the given…

Q: Prove that for all a, b, and c e Z, if a † (2bc+12), then a † (b – 2) or a ł (c + 3).

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Q: Let Z1, Z2,..., Zn be complex numbers such that Re zk> 0 and Re(zi... Zk) > 0 for 1 ≤ k ≤n. Show…

A: Given: Let consider  z1, z2..., zn be complex numbers. such that Re zk > 0 and Re(z1...zk) > 0…

Q: Show that there exist m and k such that ma + kn = 1if gcd(a, n) = 1

A: Let gcda,n=1.

Q: In Z[i], show that 3 is irreducible but 2 and 5 are not

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Q: Show that the characteristic of Z[i]/<a + bi> divides a2 +b2.

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Q: For allne z, gcdin,nt)=l.

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Q: Theorem 4.17. Let m, n be two odd positive integers such that (m, n) = 1. Then we have () () = (-¹)…

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Q: Prove that gcd(k; n) = gcd(n - k,n).

A: Greatest common divisor (gcd) :            If a , b are two integers such that atleast one different…

Q: Show that x and 2 are coprime in Z[x]

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Q: Prove that for a E N, a always invertible in Z,2+3a+1·

A: We know that an element a∈ℤn is called invertible if there is an element b∈ℤn such that a+b≡0  mod…

Q: Prove that if ACØ then A = (Ø.

A: To prove the given statement.

Q: Prove that, IXI<a Rf and If -aL X Z a

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Q: A fossilized bone is found to contain one-thousandth of the C-14 level found in living matter.…

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Q: 2. List all elements of A = M₂ (Z₂). Give the units in A, as well as the zero divisors in A.

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Q: prove that v<e+1 must contin a cycle

A: Let G be a graph with number of vertices, v and number edges, e. To prove that if v&lt;e+1, then G…

Q: Prove, without using any results from the chapter notes, that for all z = C, z² = 2².

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Q: Prove if a e G, o(a) =n, then a = reN GCD(n,r)'

A: Given below the detailed solution

Q: Prove Z(60) is cyclic

A: To show that Z(60) is cyclic. It is enough to show that there is a generator which generates Z(60).…

Q: Show that φ(n2) = nφ(n) for every n ∈ Z+.

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Q: Prove directly (do not use Theorem 1.8) that, if {an} ∞n=1 and {bn} ∞n=1 are Cauchy, so is {an + bn}…

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Q: Gillian wants to create a mixed nut blend from peanuts, cashews and macadamia nuts. The peanuts cost…

A: According to the question:   The cost of peanuts = $2 per pound   The cost of cashews = $4 per pound…

Q: Let ECR. Prove that m*(E)= 0 implies that E is Lebesgue measurable.

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