Prove that (Z,⋅,1) is a commutative monoid using the formal definition of multiplication from the video. Furthermore, prove that multiplication distributes over addition in the integers. Hint: You first need to show that given u,v ∈N×N, d(uv) = d(u)d(v) (this is similar to how, in the video, we proved d(u+v) = d(u) + d(v)). Doing this will make your life dramatically

Q: 1. Consider the set V of all polynomials of degree two or less, along with the following definitions…

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Q: التاريخ : 2 We can rewrite axt(1-d)y-xo. as follows @q (x-x₂) + (-a)y 2 (x-y)- (X-X₁) a(x-X ₂) +…

A: Using the property of inner product. In question #2 , if the metric is induced by inner product then…

Q: Let a=⟨3,−5,5⟩ and b=⟨−1,5,2⟩ Find the component of b onto a

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Q: Let 2 = {a, b, c}. Write down all possible o-algebras on N.

A: Given: Ω=a,b,c

Q: 3. Show that Z10 is isomorphic to Z2 × Z5, but that Zg is not isomorphic to Z₂ × Z4.

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Q: Is this statement true? If Ax = b has a solution for b, then A must be invertible.

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Q: Зх — Зу — 6z %3D —3 2х — 2у — 4z %3D 10 2х + Зу + z %3D7 -

A: Given system of equations, 3 x-3 y-6 z=-32 x-2 y-4 z=102 x+3 y+z=7 The objective is to solve the…

Q: Prove that if a eZ then the only positive divisor of both a and a+1 is 1.

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Q: Z3-22 If z1 = -4 - j5, Z2 = -3 + j2 , and z3 = 2 - j3, what is Re Note: z* is the complex conjugate…

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Q: -8 -6 - 41 | + 3. 10 -6] Compute: 4 · 1

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Q: (15) Let a, b = Z. Prove that if a is divisible by 4 and b is divisible by 4, then: (a) ab is…

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Q: Compute det A, det B, det (AB), and det (A + B) for -2 1 2 +*+ A = -3 -1 -4 4 −1 3 det A = det (AB)…

A: A=-212-3-1-44-13  B=2-3-4-1-3-3217

Q: Compute det B4 where B = det B4 = 102 2 14 2 3 1 (Simplify your answer.)

A: Given: The matrix .To find .Concepts used:To find the determinant of a square matrix, pick any row…

Q: Draw triangles and circles to show each mass in kg. a)x+3 b) 2x + 1 4000 A00O A000 3(2x + 1) ΔΔΟ…

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Q: ply DeMorgan's Laws un - && (Ir || a))

A: Introduction: The formula for de morgan's law is given by, ~(a∧b)=~a∨~b~(a∨b)=~a∧~b

Q: Show that A is diagonalizable.

A: Consider a square matrix A. Then A is diagonalizable if there exists an invertible matrix P such…

Q: Suppose the multiplication cx is defined to produce (cx1, 0) instead of (cx1, cx2 ). With the usual…

A: The multiplication cx where x∈ℝ2 is defined as cx=cx1, 0 because x=x1, x2 where x1 and x2 belong to…

Q: 6 Find product A) 3 -4 7 13 -9 5 ㅋ 8 of AB

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Q: The multiplicative inverse of 1+ 4x in Zg[x] is: 4x+1 6x+1 None 1/(3x+1)

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Q: Please solve this linear algebra question. Thank you

A: Consider the provided question, Let V be a vector space over F.

Q: Find, if possible, A + B, A- B, 2A, 2A- B, and B + A = 5-6 2 3 -4 1 (a) A + B 183 11 (b) A - B 188…

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Q: Determine the degree of the splitting field of x4 3 over Q. Justify your - answer.

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Q: 12. Which of the following polynomials is reducible over Q а) 4x3 + x — 2 b) Зх3 — 6х? +х - 2 c) 5x3…

A: Reducible, Irresistible

Q: 2. Prove that if 5BA+ ACA – I = A, then A must be invertible.

A: We have a solve the problem

Q: Find the integer (a+b) with respect to the module 2,3,5 if a=11 and b=13

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Q: -2 3 If A = Then, A-1 = %3D -6 4 4 2 a) 최 ( 6 3) 4 -2 b) – 6. 3 4 c) 24 6 -3 d) The inverse is not…

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Q: (c) How many of the elements of Z186 have multiplicative inverses?

A: To find the number of elements in the set of integers modulo 186, which have multiplicative inverse.

Q: The conjugate of (a-b) is (a+b) given that a² - b² = (a-b)(a+b) a. True  b. False

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Q: 3. Is B the inverse of A? [17 25 ‚B = -2 3 -25 (d) A = 17 [17 25] 24 36 3 (e) A= %3D -2 12

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Q: a) Factor a* + a3 +x into irreducible polynomials in Za[z]; b) Find the multiplicative inverse of…

A: The multiplicative inverse of (x2+1) in given ring is square (x2-1). For justification locate step 2…

Q: În a ring R, show that i) Oa = a0 = 0 ii) a(- b) = (- a)b=- (ab).

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Q: find AB, BA, A +B, A - B, A, B, 5A, 3B. Observe that AB + BA. Show that (A - B)(A + B) # (A + B)(A –…

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Q: 10. When dividing x* + 3x? + 2x + 1 by x² + 2x + 3 in Z5[x], the remainder is а) 0 b) 2 c) 3 d) 4 a)…

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Q: Showthat S subring of RIKRZ+R3

A: For the solution of the problem follow the next step.

Q: Find the matrix of the quadratic form. Assume x is in R². 9x² - 12x₁x₂ + 8x²2 b. 8x² +10x₁x2

A: The quadratic form:a) b)  To Find:Matrix of quadratic form.

Q: 1) х*- 27 2 - A) (x – 3) B) (x- 3)(x² + 3x + 9) C) (x+ 3)(x² – 3x + 9) D) (x – 3)(x + 3)

A: Due to Bartleby guidelines, only 1question can be solved. Please repost the remaining questions to…

Q: If GCD of (a,b) is 1 we can find the lcm by multiplying a and b. True or false

A: if GCD(a,b) is 1, We can find lcm by multiplying a and b.   This statement is True.

Q: 1+xr 1+x* a) b)

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Q: Show that the system consisting of all elements of the form a+b√5, where a and b are any rational…

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