P7 Given a set 2i and a measurable space (S2, F2), let f : N1 → 2. Fi = {f-'(B) : B e F2}. (1) %3D Show that Fi is a o-field and that f is measurable with respect to Fi and F2. Such F1 is called the o-field generated by f and it is denoted by o(f).

Q: 7. Prove that the function f:Z → Z defined by f (x) = x + 8 is one-to-one and onto.

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Q: Show that the operator U is positive U: 1² ([0,1]) → L²([0,1]), Uƒ (x) = X[{0,1] [~*\f(t)\dt

A: To show that the operator U is positive, need to demonstrate that for any f∈L2([0,1]), the inner…

Q: be a field and let c,d e F. Show that c · (-d) = -(c d).

A: Associative Property of Field F for a,b,c∈F a·b·c=a·b·c

Q: Let f:R → R such that f(x) = x² – 2x + 1. Verify whether f is bijective ? explain your answer.

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Q: Let V = span{e2", xe2", x²e?=}. (a) Show that (a1e2a + a2xe2" + a3x²e2") e V for any a1, a2, az E R,…

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Q: 5. Let f be defined for all real numbers and suppose f(x)-f(y)| ≤ (x - y)² for all real numbers, x…

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Q: Let C be the triangle with vertices. l6,0), (22,0), aed Ge,25) traveling Coaterdlehmle' let F be the…

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Q: 1.T.2 Let V be the set of all twice-differentiable functions f : R → R satisfying f"(x) = f(x) – 5.…

A: Let, f and g be two functions in V.  Then,        f"(x) = f(x) - 5       g"(x) = g(x) - 5.

Q: Test the field f to determine if it's conservative. F=x⁵y⁵z⁵i+x⁵y⁵z⁵j+z⁵k Conservative or not…

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Q: Show that Vx (V.f) = 0 for any vector field f.

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Q: #24 cal Show that f: IR→IR, x-> (1-x)+cos(1+x³) (a) is continous. ay Show that the map 2 f. IR→IR ×…

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Q: lee f: R>R be a C function ミく。 proper tg with 2f show that F is cunservative

A: Please check step 2 for the solution.!

Q: If F = ⟨F1, F2, F3⟩ what is div (curl(F))

A: F = ⟨F1, F2, F3⟩

Q: Suppose F = (2y, -2x). Complete the following table of values of F. Values of F y = 1 y=0 y = -1 x =…

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Q: If f is holomorphic on the D(0, 2) and f(x) = x4 – 2x² for all x = [0, 0.2], show that f(i) = 3.

A: Given that f is holomorphic on the D0,2 and f(x) = x4-2x2 for all x ∈ 0,0.2