Let J = (0, 1), the open unit interval. Define TC P(J) as follows:T= {0, J}U {(0, 1-1): neN\{1}}1. Verify that T is a topology on J.In (J, T), evaluate, without proof, each of the following:int((0, 1), (0, 1], int((,1001), (0.9999In (J, T), evaluate C(1)neN and prove your assertion.it is normal3

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Answered: Let J (0, 1), the open unit T= {0, J}U…

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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Q4. (a) Let > be a o- Algebra on X. Assume that µ1, H2 : ), – [0, ∞0) are two measures on X. Is µ1 + ku, a measure on X? where k is a scalar. Consider the cases: k >0 and k < 0. (b) Let (X, ),) be a measure space. For A , Be ). Show that μ(AU B ) μ (Α) + μ (Β) - μ(Αn Β).
How large should n be to guarantee that the Trapezoidal Rule approximation to 14x – 60z? – læ + 3)dx is accurate to within 0.1. 5 n = How large should n be to guarantee that the Simpsons Rule approximation to | (-24 – 14x – 60z? – læ + 3)dx is accurate to within 0.1. n = Hint: Remember your answers should be a whole numbers, and Simpson's Rule requires even values for n
Recall the inner product (f,g) = S, f(x)g(x) dx defined on the vector space C([-]) of continuous (real-valued) functions f : [-, π] → R. Observe the following integral formulas, which are valid for natural numbers m, n > 1: • sin(ma) cos(nx) dx = 0 (0 cos(mx) cos(nx) dx = sin(mx) sin(nx) dx = ifm #n | if m=n 1. What do these formulas say about the angles between the vectors sin(n.), sin(m.), cos(n.) and cos(m.) in the space C([-])? What do say you about the lengths of these vectors? 2. Do you think these vectors form a basis for the space C([-,])? Why or why not?
1) If CI(A)= Int B = (1,4) for some subsets A,B of [0,6].Then ([0,6],T) is: A)disconnected B) connected C) second countable D)separable 2) The following is a first countable space: A) (R, Tcot) B) ([0,1],Tcoc) C) (R,Tdis) D) (R-Q, Tcot) 3) The following is a second countable space: A) (R, Tcot) B) (Q ,Tcot) C) (R,Tdis) D) (R-Q, Tcoc)
TF Fis Riemann integrable on [arb] Hhen show IFI is Riemann un integrable [aib]
(- Let R be a collection of Riemann integrable functions on [a, b], then Prove the following: Iff ER then Ifl € R and f² € R The converse of (1) is not true. 1) 2)
FIND THE ADHERENT OF A, CLOSURE OF A, INT(A), EXT(A), FR(A)
7) Compute the divergence and the curl of V = x² i + y²j+z²k.
1) If CI(A)= Int B = (1,4) for some subsets A,B of [0,6].Then ([0,6],T) is: A)disconnected B) connected C) second countable D)separable 2) The following is a first countable space: A) (R, Tcot) B) ([0,1],Tcoc) C) (R,Tdis) D) (R-Q, Tcot) 3) The following is a second countable space: A) (R, Tcof) B) (Q ,Tcot) C) (R,Tdis) D) (R-Q, Tcoc)
S(A)= Set of all f: A-A which is both one to one and onto. plz with details
Lemma 60: If a ∈ Z then a · 1 = a. Exercise 28: Prove Lemma 60.

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Let J (0, 1), the open unit T= {0, J}U {(0, 1-1): ne N\{1} 1. Verify that T is a topology on J. 2 In (J, T), evaluate, without proof, each of the following: int((0, 1), (0, 1], int((1, 1), (1,1]. } 99 99 100 100 -)neN and prove your assertion. 3 In (J, T), evaluate C( n+1