Let G = R – {-1}. Define the binary operation * on G by x * y = x +y+xy.a) Show that (G, *) is an abelian group.b) Let H = R – {0}. Show that (G, *) and (H, ·) are isomorphic where · represents the usual product on R.

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Answered: Let G = R – {-1}. Define the binary…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Define * on Z by r * y = x + y + 4. Then: (a) Prove that * is a binary operation on Z. (b) Show that * is an associative operation on Z. (c) Determine the identity element for * in Z. (d) Show that every element of Z is invertible with respect to *. (e) Prove that (Z, *) is an abelian group.
II. Let G₂ be the solid above the xy-plane that is inside the sphere x² + y² + z² = 16 but outside the cylinder x² + y² = 12. Evaluate the triple integral using spherical coordinates. JSS ² G₂ z dV
8. Let the set G ZXZ and the binary operation on G be given by the rule (a, 6) • (c, d) = (a + r, 6+ d). It is easily verified that the pair (G, ) is a commutative group. a) Show that the mapping f: G- Z defined by fl(a, b)] = a is a homomorphism from (G, ) onto the group (2,+). b) Determine the kernel of this mapping. c) If II = {(a, a) | a E Z}, prove that (H, ) is a subgroup of (G, ), which is isomorphic to (Z,+) under the function f.
Let (IR,+) be a group of real numbers under addition and (R+,-) be the group of positive real numbers under multiplication. Prove f: R→ R+ by f (x)= ex for all x ER is homomorphism and isomorphism.
3a • Use the Bisection method to find solutions accurate to within 10-2 for x³ – 7x2 + 14x – 6 = 0 on [0,1]. Solution: Let f(x) = x³ – 7x² + 14x – 6 = 0. Note that f (0) = -6 0, therefore, based on the Intermediate Value Theorem, since f is continuous, there is p e (0, 1) such that f(p) = 0. Let ao = 0, bo = 1, with f(ao) 0. Let po = ao + bn-an sign as f(ao), therefore a1 = Po = 0.5, b1 = This yields the following results for p, and f(pn): 0.5, and we have f(po) = -0.6250 < 0 (the same bo = 1 and repeat: P1 || 2 = 0.75, ... Pn f(Pn) 0.5 -0.6250000 1 0.75000000 +0.9843750 2 0.62500000 +0.2597656 3 0.56250000 -0.1618652 4 0.59375000 +0.0540466 5 0.57812500 -0.0526237 6 0.58593750 +0.0010313
. Prove that the set S0(2) = { : z,y } forms an abelian group with respect to multiplication.
Exercise 2.2. Let (G, *) be a group satisfying that (a* b)² = a² * 62 for every a, b = G. Show that G is abelian. Extended quesion* (Opt): What happens if the condition is replaced by (a + b)³ = a³ * 6³?
Suppose that fG G such that f(x) = axa. Then fis a group homomorphism if and only if O a^3 = e a^2 e a = e a^4 = e
9. a) find two integers x,y such that 30x+101y=1. (hint: gcd(30,101)=1). b) find the inverse of 30 in the group U(101).
Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x Y of X and Y, but can we also define a binary operation on X x Y such that X x Y is a group? Let's consider two elements (x1, 41) and (x2, Y2) in X x Y. Let denote a function on X x Y.We need to define so that (X x Y,•) is a group. The most natural way to proceed is to define • component-wise: (T1, Yı) • (x2, Y2) = (x1 º Y1, ¤2 * Y2) %3D Now we should show that (X × Y,•) is in fact a group.

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Let G = R – {-1}. Define the binary operation * on G by x * y = x +y+xy. a) Show that (G, *) is an abelian group. b) Let H = R – {0}. Show that (G, *) and (H, ·) are isomorphic where · represents the usual product o

Let G = R – {-1}. Define the binary operation * on G by x * y = x +y+xy. a) Show that (G, ) is an abelian group. b) Let H = R – {0}. Show that (G, ) and (H, ·) are isomorphic where · represents the usual product o