i) Repeat the Cantor construction starting with the interval [0,1]. This time, however, remove the open “middle fourth” from thecenter of the remaining closed intervals to obtain C¹/4.i-1) Describe and draw the first 3 iterative steps of the construction of C¹/4 in detail.i-2) Using the argument above or from Section 11.1 of the book (pages 330-332), compute the length of this Cantor-like setC¹/4, and prove that C'¹/4 contains no intervals.

The cantor construction is to be done on the interval 0,1. To remove the middle fourth from the…

Answered: i) Repeat the Cantor construction…

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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For any two rational numbers a,b ∈ Q, we define a new operation a*b=a+b+ab. (a) Is Q closed under the operation *? (b) Try Prove that for any a,b,c ∈ Q, a* (b* c) = (a*b) * c. (c) Try to find e∈Q such that for any a ∈ Q, e*a = a *e= a. (d) Try to find the largest subset of Q such that it is a group under the operation of *.
Take the Set Q of the rationals and the family of intervals {(a, b)} such that a, b E Q with a <b. Where (a, b) = {x E Q // a <x <b}. show that this family meets the base conditions.
B. We extend from Q20 to Q by another application of the method of ordered pairs, exactly as we did to extend from N to Z. Thus, elements of Q can be written as ordered pairs of ordered pairs - we might for example write 1/2 as ((1,2),(0,1)). Explain how to construct the standard ordering < on the rational numbers in terms of the coordinates in these ordered pairs.
(6) Recall that a subset A of RP is dense in RP if Ã = RP. (b) Use Newton's method to construct a sequence of rational numbers con- verging to √2. Google "Newton's method", or consult a reference, if you are unsure what Newton's method is.
(ii) Find the rank and nullity of the [2 0 4 -7 2 4 6 A |1 -5 2 [7 -9 2 -4] 4 Verify that values obtained verify the dimension theorem.
[D] Consider the function F: {1,5, 8} × {6, 10} → {0, 1, 2, 3, 4, 5} defined as follows: F: (a, b) → (4a + b)mod 6 (D.1) Fill in the ordered pairs in the diagram below to indicate the elements in the domain of F. Then draw in arrows to complete the diagram so that it correspondes to the function. ( • 0 • 1 • 2 • 3 • 4 • 5 (D.2) What is the range of F? (D.3) What is the co-domain of F? 3/5 (D.4) What is the preimage (inverse image) of 3? (D.5) What is the preimage (inverse image) of 0? (D.6) Is F one-to-one? If not provide a counterexample. (D.7) Is F onto? If not provide a counterexample.
What kind of functions is the following funetion f(x) ? a) Even b) Odd c) Neither odd nor even d) None of them a b O d O e f O g O
2) Let f be the rational counting function : Solve for integer f(n) = f(5) + f (3)
Use the Least Upper Bunnd Axiom together with the usual manipulation) of algebrac idontities and inequalities to prove the following:
Let f(n) = 5" – 4. (a) Evaluate f when n is 1, 2, 3, 4, and 5. (b) For what positive integers n is f(n) divisible by 3? Prove that your guess is correct. (c) For what positive integers n is f(n) divisible by 21? Prove that your guess is correct.
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