(8) In the ring (Zg, +66), we get n {(a): a prime element in Z,} = ...... (a) (b) ((0), +66) (c) (Z6, +6. 6) (d) No Choice (a) (b) (c) (d)

Q: Prove the following: (a) If a, b,c € Z+ and a+b=c then (c, [a, b]) = (a, b).

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: a) Find C and k that show 8x? + 5× is O(x?). Show your algebra reasons. b) Use proof by…

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Q: (c) 2, 3, 4+ V10 and 4. V10 are irreducible elements of R.

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Q: Prove by induction that 1 (1)4 + 1 1 + (4)7 (7)10 1 n (3n-2)(3n+1) 3n+1

A: Given that the statement 114+147+1710+...+13n-23n+1=n3n+1Let Sn=114+147+1710+...+13n-23n+1=n3n+1

Q: Is g = {(x, y) ∈ [−5, 5] × R : x^2 + y^2 = 25} a function from [−5, 5] to R.

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Q: Find the Cholesky factorization A-[-+-+4] -1 20) A =

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Q: Show that 1+2+3+… +n = (";").

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Q: the expansion of omial theorem to determine the value of the consta 3 < (22²7³. 23 2 (2-3r³y) x…

A: Given expansion is 2-3x8y×2x2y3+3x3-2y219…

Q: 2. Determine the equation of the following function. State your answer as a piecewise function. y 10…

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Q: A = 2 B = 0 4 Compute the following if defined: (a) 3A – B (b) 2BT +C (c) AB (d) AC

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

Q: 2. Let Z [√2] = { a+b√2 la,be Z}. Define addition and multiplication on Z[√2] as follows: (a+b√2)+(c…

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Q: (6) The ring (Z,12, +12,·12 ) is ...... (a) field (b) semi-simple (c) simple (d) No Choice O (a) O…

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Q: Use the permutation and combination formulas to prove the following. +k – 1) (n + k – 1 (b). V k, n…

A: We have to  prove given formula using peemutation and combination

Q: Use a combinatorial argument to prove that EC(n, k) = 2". k=0

A: The Multiplication Principle If one event can occur in m ways and a second event can occur in n ways…

Q: Write the fourth term of the expansion of (2x+ 3y)°.

A: We can get desired result by two ways. One is by opening binomial expansion then getting fourth…

Q: What is the rule for this pattern? (figure 1 is 1 block) Of(x)=n+2 Of(x) = 2n Of(x)=n-1 Of(x) = 2n-1

A:

Q: 13 - What is Taylor polynomial with order 3 generated by f(x) = x° + 3x² + 2x + 4 at x = 1.

A: The Taylor polynomial of order 3 is given as follows :

Q: Give a combinatorial proof of the identity (C) ) = (O).

A:

Q: (a). Show that the addition of x²+x and x²+x+1 is in GF(2³) (b).Show that the multiplication of x²+x…

A: We know that GF(2m) are the binary polynomials, i.e. polynomial whose coefficients are either 0 or…

Q: Consider the matrices: 7 A = B = С — -6 6. D = [6 8 3], E = | 5 3 6. -6 3 F =| 3 9 G = 3 -3 5 3 7 5

A: In this question given that any type of metrix and help of these metrix we find the value of any…

Q: Let a ER which expressions are correct? and give the justification why they are part of the Borel…

A: Consider the given information: Let a∈ℝ  Then which statements are correct. Here Second option is…

Q: In Z[2√7], show that N(6 + 2√-7) 5 N(1 + 3√-7) but 6 + 2√-7 and 1 + 3√-7 are not associates.

A: As per the question we are given the extended field ℤ [2√-7] and we have to prove that N(6 + 2√-7) =…

Q: a) If q(n) = byn* + Bk-1n*-1+...+ b1n+ bo is a polynomial in the variable ne N with be # 0, show…

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Q: (c) Suppose n EN is divisible by 5. What is (25n) (n) L?

A: Let, the prime factorization of n be as below.          n = p1a1 p2a2 . . . . . . pkak     for some…

Q: (n² + 6n – 4)(2n – 4)

A: The polynomial is given by n2+6n-4(2n-4)

Q: Let (M) be the statement P(n), n 2 1 where P(n) is 2[1+3+32 + ·.+ 3"] = 3n+1 – 1. (M) is a false…

A: For n=1, L.H.S. = 2[1]=2.

Q: Prove (without using induction) F1- F = Fn-1Fn+2•

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Q: In the characteristic polynomial (3 – A)(4 – A)³(–3- )(3 – 1), what is the multiplicity of d = -3?

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Q: Prove that: n n (3) + (-²)=(-+ m m+1 n+1 m+1

A: We have to prove the given identity

Q: The characteristic of the ring 6x8 ×; is 14 (A) 2 (B) 48 (C) 84 (D) 168

A: We have to find the characteristic of the ringWe know thatCharacteristic of a Ring : Let R be a…

Q: 6 (la) Compute the following, recording only the answers on this page and showing all computations…

A: Given A = a0d0b000c Then A2 = a0d0b000c2= a0d0b000ca0d0b000c = a20ad+dc0b2000c2 Then A3 =…

Q: Given w(x) 5a +6 and z(x) = 3x – 4, evaluate the following. %3D %3D (a) (w o z)( – 1) = %3D (b) (z o…

A: Consider the given information. wx=5x+6zx=3x2−4 Find the composition of the function as,…

Q: Provide a combinatorial proof (argue that both sides count the same thing) for the following:…

A: Please see the below picture for detailed solution.

Q: n (3n–1) 2 B. Use mathematical induction to prove that P(n): 1+4+7+..+(3n-2) = n2 1. Use the first 2…

A: