10. Let (G, *) be a group, and let H≤ G. Define N(H) = {x € G: x¹ *H* x = H} [Normalizer of H in G]. Show that (a). N(H) ≤G. (b). HAN(H). (c). N(H) = Gif and only if HG. =

Q: e* – 1 lim V1+x -1 x→0

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Q: Let f be an arithmetical function such that f(n) > 0 for all n. Let a(n) be arithmetical function…

A: Given f be an arithmetical function such that f(n)>0 for all n. Let a(n) be arithmetical function…

Q: 2.11. Let f e r(V), V a variety c A". Define 1 G(f) = {(a1,..., An, An+1) € A"+ |(a1,..., an) E V…

A: If J is any ideal in k[x1, . . . , xn] ,we define Affine Variety as V(J):={ p∈An | f(p)=0 for all…

Q: Suppose g [a, b] → R is a continuous function that takes only irrational values. Prove : that 9 is…

A: Use the Intermediate Value Theorem (IVT) and method of contradicts.

Q: 45. Let g(x) = [x]. Find a) g-¹({0}). c) g-¹({x|0<x< 1}). b) g-¹({-1, 0, 1)).

A:

Q: An RC circuit with a 100 Q resister and a 10^(-2) F capacitor is driven by a voltage…

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Q: Suppose X and Y are r.v.s such that E(X) = 4, Var(X) = 48, E(Y ) = 2, andVar(Y ) = 5. Find upper and…

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Q: Q. Let g(x) : [2, 3] -+ R be a continuous mapping. 3 And / (9(x) + 2 – 1)'dx = 0, then - a.g(x) = 0.…

A: Given function g(x) is continuous on the interval [2,3]

Q: Let g(x) = e² · (x² – 9), g' (x) = e" . (x² + 2x – 9), and g" (x) = e² · (x² + 4x – 7) , Choose one…

A:

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A: What is Lyapunov Function: Lyapunov functions are those function satisfying the Lyapunov stability…

Q: Exercise 6. Let f(x) = 2x-5logx. Then f(x) is а. О(х) b. O(x²) с. О(Зx?) d. All of these

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Q: 24. Let f be an arithmetical function (not ne multiplicative arithmetical function g s S fl(k,n)) =…

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Q: Q3. (a) Let X = {m, n, P, q}, and = {0, X, {m}, {n}, {p, q}, {m, n}, {n,p, q}, {m, p, q}}. Is > a…

A:

Q: 6. Let (a, b, c) and let Σ* be the set of all words w using letters from Σ; see Example 2(b). Define…

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A: f(x) = ln(ex-3)The function "ln" is defined for positive numbers only.Hence, x should be such that…

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A: The objective is to solve the above question.

Q: 15.K. Let f be monotone increașing on / [а, б] to R. If c € (a, 6), dehne l(c) l(c) = sup {f(x): I…

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Q: 6. Let E≤X where X is a metric space. Show the limit points of E are the same as the limit points of…

A: Answer:-

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A:

Q: 2. Let (r) be the function defined by V(x) = 1. Let 1+x. In(x) Vovo...(x) n-copies Note that…

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Q: 10. a) What transformation are used to turn the parent function f(x) = x³ into g(x) : -(2³ – 3) +2.…

A: Given query is to find the transformation applied and then graph the function.

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Q: Show that cl(a) = {a} if and only if a ∈Z(G).

A:

Q: 4

A: We use theorem of order of group and order of elements of group .

Q: 10. |G/H| [G:H]. *True False

A: 10. True or False: |G/H|=[G:H]

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Q: 3.1.5. (-) Prove that a (G) > AG)+1 n(G) for every graph G.

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Q: LetX={0,1,2}, and consider the collection of functionsf:X→Xsuch thatfis invertible. How many…

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Q: 2.5. Give an example of a bounded function f and an increasing function a defined on [a, b] such…

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Q: 1. Let X be the graph of f(x) that is, X is the subset of R x R satisfying the given equation.

A: Sol