2.28 let {Ei} i Ej be a Measurable pairwise dis joint sets. prove that for any bounded set A countable family of M² (ANU)) = {_ M² (ANME:) iєJ ej

Q: Hwi Let A () Find all eigonvotveo of A Gnd their corispondhing eigevenvedters

A: Topic:- matrix

Q: ercise 6.67. Let E be a subset of R having at least two points. Pro- owing are equivalent:

A:

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A:

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A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

Q: 1 - Let A = {x € R: -2-4<x<5+0} B = {x € R: x²(x+y+1)x < 0} Find the set AUB,ANB, A\B, B\A A', B' as…

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A: See solution below. Here we used definition of bounded and closed sets.

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Q: 3.2 Prove that in any metric space (S.) every closed ball Se[xo] is a closed set.

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Q: 1.2 Now consider the following sets: X {(x, y) : x² + y² < 4} (5) %3D Y = {(x, y) : -x +y < -2} (6)…

A: Given: X = x, y : x2 + y2 &lt; 4Y = x, y : -x + y &lt; -2

Q: 9. Given an infinite collection An, n=1,2,... of intervals of the real line, their intersection is…

A: Let An : = [ n ,∞)  where  n = 1,2, 3 .....  . So { An } n∈ℕ is an example of infinite collection…

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Q: Exercises for Sets and Functions Q1) How many elements does each of these sets have? b. P(Ø, {Ø),{Ø,…

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A:

Q: Let A CR be measurable with m(A) > 0. Prove the following statements: There is a non-measurable set…

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A: Given N :R^n→R be a norm on R^n.      To show : N is a convex.     Definition of norm :…

Q: d) Suppose X = {1,2, 3} and Y = {5,6}. The number of elements of the set P(X× Y) =

A: Elements in X = 3. Elements in Y = 2 Elements in X × Y = 3 × 2 = 6

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Q: 23. Let A be a non-empty set of positive numbers, which is bounded above and let B = {xe R: 1/xe A}.…

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Q: Q.No.3 a) Prove that if A and B are sets, then (An B)U (ANB') = A

A:

Q: Let A CR be measurable with m(A) > 0. Prove that there is a non-measurable set NCA

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

A:

Q: b Let A be a continuous fuzzy quantity. If all the a-cuts of A are closed intervals then show that A…

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Q: Ex 4.) Let Sbe a non empty bounded set in 1. aj Let azo, and let a S:= {as: SES}. Prove that inf…

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A: We have to find the linear operator L with null space(Kernel) nL=excos2x,exsin2x

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

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Q: Let D be a Euclidean domain with measure d. Show that if a and bare associates in D, then d(a)…

A: We have to show d(a) =d(b) if a and b are associates in D

Q: 4. Assume that S, T C R are two non-empty bounded sets. Prove that: (i) SUT is a bounded set; (ii)…

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Q: (2) If SC R" prove that the collection of isolated points of S is count- able (hint: use a covering…

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A: (.) Given ,  We have to find sets  and  defined by,

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A: Consider A and B are bounded sets.

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Q: 3.17 Let A be a subset of R, and let f be a bounded function with domain A. Show that, if BC A, then…

A: Given that A,B are subsets of ℝ andB⊂A. As f is a bounded function in A and B⊂A then fB⊂fA

Q: 5)Evaluate F - nds for F = y²zi + y°j+ xzk using Gauss' Theorem. Consider that S is the boundary of…

A: We will find out the required value.

Q: Let F and G be nonempty bounded sets. Prove that inf (F UG) = min{inf F, inf G} and sup (F UG) =…

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Q: -[2.16] Let S be a nonempty set in R" and let x e S. Consider the set C = {y : y 2(х - х), A20, х E…

A: Given- Let S be a nonempty set in Rn and let x¯ ∈ S. Consider the set C = y : y = λx - x¯, λ ≥ 0, x…

Q: Let U CR. Prove U is open if and only if U = U°.

A: We have to solve given problem:

Q: c) given an arbitrary convex function wcx) :R"→tR, prove fhat is a convex set.

A: