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Suppose that G,H are groups with identities eG,eH, respectively. Define f: G x H→G by setting, for all g in G and h in H, f(g,h) = g. a) Prove that f is a homomorphism. b) Prove that f is surjective. c) Prove that ker(f) = {(eG,h) | h in H}.
(5) Let (. A, /u) be a measure space. Let f,g : O > R* be a pair functions. Assume that f is measurable and that f = g almost everywhere. (a) Prove that q is measurable on A. Prove that g is integrable (b) Let A E A and assume that f is integrable on A and A
(5) Let (. A, /u) be a measure space. Let f,g : O > R* be a pair functions. Assume that...
2) (12) f:R-(3/2)-R-10, (x) 1/(3 2x) g:R--21->R-1o), g (x)1/ (x 2) h:R-(-4/3]-R-(1/3), h(x) (f o g) (x) Verify if h(x) is one to one and onto. If it is, find the inverse function of h(x).
2) (12) f:R-(3/2)-R-10, (x) 1/(3 2x) g:R--21->R-1o), g (x)1/ (x 2) h:R-(-4/3]-R-(1/3), h(x) (f o g) (x) Verify if h(x) is one to one and onto. If it is, find the inverse function of h(x).
Prove that (f = O(g)] ^ (g = O(h)] = f = O(h).
number 1 and 2 pls
Problem 1.1. Suppose that f: R → R and that f is differentiable at z = a. 1. Show that, given an angle 6, we can choose 6(0) > 0 small enough so that for all r such that r - al < (0) we have that the graph of f(r) lies inside of the cone with angle e around the tangent line. 2. Can you find explicit formulas for 6(0) for the function f(x)...
Let U ⊆ R^n be open (not necessarily bounded), let f, g : U → R
be continuous, and suppose that |f(x)| ≤ g(x) for all x ∈ U. Show
that if
exists, then so does
.
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H=1.68 T=26 F=2 L=2 S=30
11) Determine the extrema of g(u,v) = Hu-(F+L)t/2 subject to the constraint x2+y2= S*H Ans.
J-J, f(x)--3, g : S → J, g(s) = nuniber of elements in the string 's', if is even. h : J-J, h(1)- r r if is odd - . where J denotes the set of integers and S denotes the set of all character strings. Calculate each of the following if they exist (if they do not exist explain why they do not exkt): (i) fo r (i) ho f(x), 8 marks) (ii) hofo g(test)
J-J, f(x)--3, g :...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
(8) Let E C R" and G C R" be open. Suppose that f E G and g G R', so that h = go f : E → R. Prove that if f is differentiable at a point x E E, and if g is differentiable at f (x) E G, then the partial derivatives Dihj(x) exist, for all and j - ...., and 7m に! (The subscripts hi. g. fk denote the coordinates of the functions h, g....