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![Exercise 10.3. Given two rings R₁1 and R2, show that (R₁ × R₂)[x] and (R₁[x]) × (R₂[x]) are
isomorphic (as rings).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2e948f6-fd6f-485f-942e-c931230f8579%2Fbf1ee082-d040-4427-adb9-90cebde417d3%2Fmeyn18c_processed.jpeg&w=3840&q=75)
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- Suppose that Pr[A|B]=713, Pr[B|A]=711, and Pr[A′]=1021. Find Pr[A∩B′].In the frieze group F7, show that zxz = x-1.b) Find Af(x) if f(x) = (3x+1)(3x+ 4)(3x+7)... (3x+ 19). / c) Let A = {1,2,3}, B = {a, b, c}, and C = {x, y, z}. Consider the following relations R and S from A to B and from B to C, respectively. R = {(1,b), (2,a), (2, c)} and S = {(a, y), (b,x), (c, y), (c, z) }. %3D %3D Find the matrix representation of S°R
- aw- Let Z[/2] = {a +b/2 |a, b e Z} and let H = { b а 2b : a,b eZ }. a Show that Z[ /2] and H are isomorphic as rings.Compute f F dr, where F(x, y) = (5 + 27x², 2xy) and C is a line segment from (0, 0) to (1.2). (Use symbolic notation and fractions where needed.) [ F. dr =Find all values of z such that z5 =1 - i.
- a.) Calculate L6 for f(x)= (squareroot (9x+1)) over [1,3] Give answer to three decimal places L6= b.) Calculate R5 for f(x)=3x^2+5x over [-1,1] (Give an exact answer. Use symbolic notation and fractions where needed.) R5= c.) Calculate M6 for f(x)=(squareroot(2x)) over [2,5] (Use decimal notation. Give your answer to two decimal places.) M6=(Q1)Find all idempotent elements and nilpotent elements of the following rings 2- (Z/(3),+.) + (C. +,.)If x and y are 3−cycles in Sn, prove that ⟨x, y⟩ is isomorphic to Z3, A4, A5 or Z3 × Z3.
