Real Analysis MSQs and MSQs : Point Set Topology - II

13. Which of the following sets satisfy the condition that for every positive integer \(n\) there is some \(a\) in \(A\) such that \(a

(a.) \(A=\{-1,5\}\)
(b.) \(A\) is the empty set
(c.) \(A=\mathbb{N}\)
(d.) \(A=\{x \in \mathbb{R} \mid x>10\}\)

Explanation for Question 13:
a) For all \(n \in \mathbb{N}\), \(-1 \in A\) such that \(-1 < n\).
c) If \(A = \mathbb{N}\), then for \(1 \in \mathbb{N}\), there does not exist \(a \in A\) such that \(a < 1\).
d) If \(A = \{x \in \mathbb{R} \mid x > 10\}\), then for \(1 \in \mathbb{N}\), there does not exist \(a \in A\).

14. Let \(S\) be an infinite subset of \(\mathbb{R}\) such that \(S \cap \mathbb{Q}=\phi\). Which of the following statements is true?

(a.) \(S\) must have a limit point which belongs to \(\mathbb{Q}\)
(b.) \(S\) must have a limit point which belongs to \(\mathbb{R} \backslash \mathbb{Q}\)
(c.) \(S\) cannot be a closed set in \(\mathbb{R}\)
(d.) \(\mathbb{R} / S\) must have a limit point which belongs to \(S\)

Explanation for Question 14:
a, b, c) Let \(S = \{\sqrt{n} \mid n \in \mathbb{N}\}\). The intersection of \(S\) with \(\mathbb{Q}\) is empty, and no limit point belongs to \(\mathbb{Q\).} Also, \(S\) is closed.
d) \(\mathbb{Q}\) is a subset of \(\mathbb{R} \setminus S\), and \(\mathbb{Q}\) is dense in \(\mathbb{R}\). Therefore, for all \(s \in S\), \(s\) is a limit point of \(\mathbb{Q}\).

15. Let \(A \neq \mathbb{R}\) be a dense subset of \(\mathbb{R}\). If \(U \subseteq \mathbb{R}\) is a non-empty open subset, then

(a.) \(U \subseteq \overline{A \cap U}\)
(b.) \(\overline{A \cap U}=\phi\)
(c.) \(\overline{A \cap U} \subseteq U\)
(d.) \(\overline{A \cap U}=A \cap \bar{U}\)

Explanation for Question 15:
Since \(A\) is dense in \(\mathbb{R}\), it is also dense in \(U\).
- \(\overline{A \cap U} = \overline{U} \), which implies \(U\) is in \(\overline{A \cap U}\).
- \(A \cap U\) is not empty.
- For \(A = \mathbb{Q}\) and \(U = (0, 1)\), \(\overline{A \cap U} = [0, 1] \subseteq (0, 1) = U\).
- For \(A = \mathbb{Q}\) and \(U = (\sqrt{2}, \sqrt{3})\), \(\overline{A \cap U} = [\sqrt{2}, \sqrt{3}]\), but \(A \cap \overline{U} \neq [\sqrt{2}, \sqrt{3}]\).

16. If \(S=\left\{\frac{1}{2m}+\frac{1}{3n}; m,n\in \mathbb{N}\right\}\), then the derived sets \(S'\) and \(S^*\) have the property

(a.) \(S'\) and \(S^*\) both are finite
(b.) \(S'\) is finite and \(S''\) is empty
(c.) \(S'\) is infinite and \(S''\) is a singleton
(d.) \(S'\) and \(S''\) both are infinite

Explanation for Question 16:
- \(S = \left\{\frac{1}{2m} + \frac{1}{3n} \mid m, n \in \mathbb{N}\right\}\). - \(S^\prime = \left\{\frac{1}{2k}, \frac{1}{3j} \mid k, j \in \mathbb{N}\right\}\). - \(S^{\prime \prime} = \{0\}\).

17. Let \(A=\left\{1+\frac{1}{n}; n\in \mathbb{N}\right\}\) and \(B=\left\{1-\frac{1}{n}; n\in \mathbb{N}\right\}\). What is the derived set of \((A\cup B)\)?

(a.) Has cardinality 2
(b.) Has cardinality 1
(c.) Is the empty set
(d.) Is a countably infinite set

Explanation for Question 17:
- \(A \cup B = \left\{1 + \frac{1}{n}, 1 - \frac{1}{m} \mid m, n \in \mathbb{N}\right\}\).
- \((A \cup B)^\prime = \{1\}\).

18. Let \(S=\{x\sqrt{2}; x\in \mathbb{Q}\}\). What is the derived set of \(S\)?

(a.) The empty set
(b.) A singleton set
(c.) Equivalent to the set of natural numbers
(d.) An uncountable set

Explanation for Question 18:
\(A = \{x \cdot a \mid x \in \mathbb{Q}\}\) for any \(a \in \mathbb{R}^*\), and it is dense in \(\mathbb{R\).

19. In \(\mathbb{R}\), let \(F_{n}=\left(\frac{-1}{n}, \frac{1}{n}\right)\) for \(n=1,2,3,\ldots\). What is \(\bigcap_{n=1}^{\infty} F_{n}\)?

(a.) The empty set
(b.) Open but not closed
(c.) Closed but not open
(d.) Both open and closed

Explanation for Question 19:
\(F_n = \left(-\frac{1}{n}, \frac{1}{n}\right)\).
\(\bigcap_{n=1}^{\infty} F_n = \{0\\).

20. If \(S\) is the finite intersection of the family of all neighborhoods of a point \(x\), then

(a.) \(S\) is an infinite set.
(b.) \(S\) is a singleton
(c.) \(S\) is finite but need not be a singleton
(d.) Cannot be determined

Explanation for Question 20:
a) \((0,1)\) is connected but not compact.
b) \(\bigcup_{n=1}^{\infty} [0, n]\) is not compact.
c) \(\bigcup_{n=1}^{\infty} \left[\frac{1}{n}, 1\right] = (0, 1]\).
d) \(\left\{\frac{1}{n} \mid n \in \mathbb{N}\right\}\) is a bounded infinite set without a limit point in itself.

21. Let \(A\) be a non-empty subset of \(\mathbb{R}\) having a supremum. Let \(B=\{x\in \mathbb{R}:\frac{(x+1)}{2}\in A\}\). What is \(\sup B\)?

(a.) \(\sup B=2 \sup A-1\)
(b.) \(\sup B=(\sup A+1)/2\)
(c.) \(\sup B=\sup A\)
(d.) \(B\) does not have a supremum

Explanation for Question 21:
\(x\) and \(\frac{x+1}{2}\) are directly proportional.
\(\sup A\) corresponds to \(\sup B\).
This implies \(\frac{\sup B+1}{2} = \sup A\).
Therefore, \(\sup B = 2\sup A - 1\).

22. Let \(A\) be any subset of \(\mathbb{R}\) such that \(A \cap A' = \emptyset\), where \(A'\) is the set of all limit points of \(A\). Then,

(a.) \(A\) is uncountable
(b.) \(A\) is finite
(c.) \(A\) is countable
(d.) None of these

Explanation for Question 22:
If \(A\) is uncountable, then \(A \cap A^{\prime}\) is uncountable, which implies \(A\) must be countable.
For example, if \(A = \mathbb{N}\), then \(A \cap A^{\prime} = \emptyset\), and \(A\) is not finite.

23. What is the supremum of the set \(S=\left\{1,1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{2^{2}}\ldots,1+\frac{1}{2}+\frac{1}{2^{2}}\ldots\frac{1}{2^{n-1}}\right\}\)?

(a.) 1
(b.) 2
(c.) 3
(d.) \(\frac{1}{2}\)

Explanation for Question 23:
Elements of \(S\) are finite series of a geometric progression (G.P.).
The limit of the sequence is \(2\), which is increasing.
\(\sup S = \text{limit of the G.P. sequence} = 2\).

24. Let \(A=\left\{x:x=\frac{4n+3}{n}; n\in\mathbb{N}\right\}\). What are \(\sup A\) and \(\inf A\) respectively?

(a.) \(\sup A=7\) and \(\inf A=4\)
(b.) \(\sup A=0\) and \(\inf A=4\)
(c.) \(\sup A=3\) and \(\inf A=4\)
(d.) None of these

Explanation for Question 24:
\(\left\langle\frac{4n+3}{n}\right\rangle = \left\langle 4+\frac{3}{n}\right\rangle\) is a decreasing sequence.
\(\Rightarrow \text{supremum} = 4+\frac{3}{1} = 7\).
\(\& \text{infimum} = 4\).