E. None of the above.21. Which of the following is/are a null set?I. {r:r is prime and 7 << 10}A. I and II onlyII. {r:r€ 0}II. {r € N: 0< 1 < 1}В. 1 onlyC. II and III onlyD. III onlyE. None of the above choices A, B, C, or D.22. Which of these statements is/are not true?I. Every decreasing sequence of positive integers has a finite number of terms.II. Every bounded sequence in R is convergent.III. Every monotonic increasing sequence of real numbers is convergent.A. I onlyB. I and II onlyС. I аnd II оnlyD. I, II and IIIE. None of the above choices A, B, C, or D.23. Find sup A if A -{ :z EIRbe a bounded subset of R.A.В. -C. 1+ v2D. 1- V2E. None of the above.K. Piesie.Page 6 of 824. Which of these statements is /are not true?I. Every bounded subset of R always contains its lower and upper bounds.II. The infimum and the supremum of any bounded subset of R are unique.III. Every monotonic decreasing sequence of real numbers that is bounded belowconverges to its infimum.A I and II onlyB. II and III onlyC. I, II and IIID. III onlyE. None of the above choices A, B, C or D.25. Let a, :-(), n>1 Which of these conditions must be met if a, is to convergeto 1?I. for any e > 0, n >.II. for any e > 0, ke vn 2 n(e)..II. for any e > 0, | >e V n 2 n(e)..A I and II onlyВ. 1 onlyC. II and III onlyD. I and III onlyE. None of the above choices A, B, C or D..26. A sequence a, is defined inductively by a = 2 and an+1 -, n2 1. If a, isdecreasing and bounded below, find the limit of a,.A. (3+ V5)B. (3+ V3)C. (3 – V5)D. V5E. None of the above.

As this is a multiple question type, only first question is to be solved, according to the Bartleby…

Answered: 21. Which of the following is/are a mll…

Math

Advanced Math

21. Which of the following is/are a mll set? 1. (r:z is prime and 7

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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Concept explainers

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

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E. None of the above.
21. Which of the following is/are a null set?
I. {r:r is prime and 7 << 10}
A. I and II only
II. {r:r€ 0}
II. {r € N: 0< 1 < 1}
В. 1 only
C. II and III only
D. III only
E. None of the above choices A, B, C, or D.
22. Which of these statements is/are not true?
I. Every decreasing sequence of positive integers has a finite number of terms.
II. Every bounded sequence in R is convergent.
III. Every monotonic increasing sequence of real numbers is convergent.
A. I only
B. I and II only
С. I аnd II оnly
D. I, II and III
E. None of the above choices A, B, C, or D.
23. Find sup A if A -{ :z EIR
be a bounded subset of R.
A.
В. -
C. 1+ v2
D. 1- V2
E. None of the above.
K. Piesie.
Page 6 of 8
24. Which of these statements is /are not true?
I. Every bounded subset of R always contains its lower and upper bounds.
II. The infimum and the supremum of any bounded subset of R are unique.
III. Every monotonic decreasing sequence of real numbers that is bounded below
converges to its infimum.
A I and II only
B. II and III only
C. I, II and III
D. III only
E. None of the above choices A, B, C or D.
25. Let a, :-
(), n>1 Which of these conditions must be met if a, is to converge
to 1?
I. for any e > 0, n >.
II. for any e > 0, ke vn 2 n(e)..
II. for any e > 0, | >e V n 2 n(e)..
A I and II only
В. 1 only
C. II and III only
D. I and III only
E. None of the above choices A, B, C or D..
26. A sequence a, is defined inductively by a = 2 and an+1 -, n2 1. If a, is
decreasing and bounded below, find the limit of a,.
A. (3+ V5)
B. (3+ V3)
C. (3 – V5)
D. V5
E. None of the above.

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