q2 7. Suppose that X and Y are disjoint sets and k E Z>o. Prove that the function:U(P:(X) × Pk-i(Y)) → Pr(X UY)i=0p(A, B) = AU Bis a bijection. Deduce that(":") -±(")(")m + nnki=0for any m, n, k e Z>o.

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Answered: 7. Suppose that X and Y are disjoint…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.3: Divisibility

Problem 30E: Let be as described in the proof of Theorem. Give a specific example of a positive element of .

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Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .
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The main point of this exercise is to use Green’s Theorem to deduce a specialcase of the change of variable formula. Let U, V ⊆ R2 be path connected open sets and letG : U → V be one-to-one and C2such that the derivate DG(u) is invertible for all u ∈ U.Let T ⊆ U be a regular region with piecewise smooth boundary, and let S = G(T). Solve A B C
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(a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by expressing u directly in terms of x, y, and z before differentiating. Then (b) evaluate ux, u y, and uz at the given point (x, y, z). u = (p - q )/(q - r ), p = x + y + z, q = x - y + z, r = x + y - z; (x, y, z) = (sqrt(3), 2, 1)
The main point of this exercise is to use Green’s Theorem to deduce a specialcase of the change of variable formula. Let U, V ⊆ R2 be path connected open sets and letG : U → V be one-to-one and C2such that the derivate DG(u) is invertible for all u ∈ U.Let T ⊆ U be a regular region with piecewise smooth boundary, and let S = G(T). Answer C

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7. Suppose that X and Y are disjoint sets and k E Z>o. Prove that the function k so:U(P:(X) × x Pk-i(Y)) → Pk(XUY) i=0 p(A, B) = AU B is a bijection. Deduce that (":") -¿(")(-) m + n n k i=0 for any m, n, k e Z>0.