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- Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.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: .
- If e is the unity in an integral domain D, prove that (e)a=a for all aD. [Type here][Type here]1. Suppose E⊆X , where X is a metric space, p is a limit point of E , f and g are complex functions on E and fx=A and gx=B . Prove fgx=AB if B≠0Let Qc(x) = x2 + c. Prove that if c < 1/4, there is a unique µ > 1 suchthat Qc is topologically conjugate to Fµ(x) = µx(1 − x) via a map of theform h(x) = αx + β.
- Given the tent map T(x) = {2x for x<= 1/22-2x for x> 1/2}Prove that the set of all periodic points of T is dense in [0, 1] and determine the number of points with least periods and their distinct orbits.this question is differential geometry Let F: IR3 →IR3 is a diffeomorphism and M is a surface in IR3 , prove that the image F(M) is also a surface in IR3.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
- Let f: S¹→ S¹ be a homeomorphism. Prove that h(f) = 0Show that the Lie bracket defined on L/I is bilinear and satisfies theaxioms (L1) and (L2) (Algebra Lie)Show that the functions f, g : D^1 → D^1, f(x) = x^2 , g(x) = 1/2sin(x) are homotopic, where D^1 is the closed unit disc in E^1 and E^1 is R equipped with euclidean topology.