Practice Questions for NET JRF Real Analysis Assignment: Countability of Sets - III

21. Consider the following statements.

1. \(S\) be the set of all straight lines in a plane, each of which passes through at least two different rational coordinates.

2. If \(S=\left\{x: x\text{ be a rational point of }\mathbb{R}^{2}\right\}\) (a point \(x=\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}\) is called a rational point if both \(x_{1}, x_{2} \in \mathbb{Q}\)), then

(a.) 1 and 2 both countable
(b.) 1 countable and 2 uncountable
(c.) 2 countable and 1 uncountable
(d.) 1 and 2 both uncountable

22. Consider the following statements.

1. If every subset of a set is countable, then the set is countable.

2. If every proper subset of a set is countable, then the set is countable.

Then

(a.) 1 correct 2 may or may not correct
(b.) 2 correct 1 may or may not correct
(c.) Both may or may not correct
(d.) Both correct

23. If \(F=\{A_1, A_2, \ldots\}\) is a countable collection of countable sets, let \(\{B_1, B_2, \ldots\}\) where \(B_1=A_1\) and for \(n>1\),

\(B_n=A_n - \bigcup_{k=1}^{n-1} A_k\).

Then form the following statements which is/are true:

1. \(G\) is a collection of disjoint sets.
2. \(\bigcup_{k=1}^{\infty} A_k = \bigcup_{k=1}^{\infty} B_k\).
3. \(\bigcup_{k=1}^{\infty} B_k\) is countable.

Code:

(a.) Only 1
(b.) 1 and 3
(c.) 2 and 3
(d.) All three

24. Which of the following is/are true?

(a.) The set \(\{e^x : x \in \mathbb{R}\}\) is a countable set
(b.) The set \(\{\log x : x > 0\}\) is a countable set
(c.) \(\{\sin x : -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\}\) is a countable set
(d.) The set of all solutions of the equation \(z^n = 1 ; n = 1,2, \ldots\), where \(z\) is a complex number, is a countable set

25. If \(A=\{(x, y): y=e^x,\, x \in \mathbb{R}\}\) and \(B=\{(x, y): y=e^{-x},\, x \in \mathbb{R}\}\) then

(a.) \(A \cap B = \emptyset\)
(b.) \(A \cap B \neq \emptyset\)
(c.) \(A \cup B = \mathbb{R}\)
(d.) None of these

26. Let \(X\) denote the two-point set \(\{0,1\}\) and write \(X_j = \{0,1\}\) for every \(j = 1,2,3, \ldots\). Let \(Y = \prod_{j=1}^{\infty} X_j\). Which of the following is/are correct?

(a.) \(Y\) is a countable set
(b.) Cardinality of \(Y\) is equal to the cardinality of \([0,1]\)
(c.) \(\bigcup_{n=1}^{\infty} \left(\prod_{j=1}^{n} X_j\right)\) is a countable set
(d.) \(Y\) is an uncountable set

27. Which of the following subsets of \(\mathbb{R}^{2}\) is/are NOT a countable set?

(a.) \(\left\{(a, b) \in \mathbb{R}^{2} \mid a \leq b\right\}\)
(b.) \(\left\{(a, b) \in \mathbb{R}^{2} \mid a+b \in \mathbb{Q}\right\}\)
(c.) \(\left\{(a, b) \in \mathbb{R}^{2} \mid a b \in \mathbb{Z}\right\}\)
(d.) \(\left\{(a, b) \in \mathbb{R}^{2} \mid a, b \in \mathbb{Q}\right\}\)

28. For each \(J = 1,2,3, \ldots\) let \(A_j\) be a finite set containing at least two distinct elements, then

(a.) \(\bigcup_{j=1}^{\infty} A_j\) is a countable set
(b.) \(\bigcup_{j=1}^{\infty} A_j\) is an uncountable set
(c.) \(\prod_{j=1}^{\infty} A_j\) is an uncountable set
(d.) \(\prod_{j=1}^{\infty} A_j\) is a countable set

29. Consider the sets of sequences:

\(X = \left\{\left(x_n\right) : x_n \in \{0,1\}, n \in \mathbb{N}\right\}\) and

\(Y = \left\{\left(x_n\right) \in X : x_n = 1 \text{for at most finitely many} n\right\}\)

Then,

(a.) \(X\) is a countable set, \(Y\) is a finite set.
(b.) \(X\) is an uncountable set, \(Y \) is a countable set.
(c.) \(X\) is a countable set, \(Y\) is a countable set.
(d.) \(X\) is an uncountable set, \(Y\) is an uncountable set.

30. Let \(A\) be any set. Let \(\mathbb{P}(A)\) be the power set of \(A\), that is, the set of all subsets of \(A\); \(\mathbb{P}(A) = \{B : B \subseteq A\}\). Then which of the following is/are true about the set \(\mathbb{P}(A)\)?

(a.) \(\mathbb{P}(A) = \emptyset\) for some \(A\).
(b.) \(\mathbb{P}(A)\) is a finite set for some \(A\).
(c.) \(\mathbb{P}(A)\) is a countable set for some \(A\).
(d.) \(\mathbb{P}(A)\) is an uncountable set for some \(A\).

31. Which of the following sets of functions are uncountable? (N stands for the set of natural numbers.)

(a.) \(\{f \mid f: \mathbb{N} \rightarrow\{1,2\}\}\)
(b.) \(\{f \mid f:\{1,2\} \rightarrow \mathbb{N}\}\)
(c.) \(\{f \mid f:\{1,2\} \rightarrow \mathbb{N}, f(1) \leq f(2)\}\)
(d.) \(\{f \mid f: \mathbb{N} \rightarrow\{1,2\}, f(1) \leq f(2)\}\)

32. Let \(A=(0,1)\) and \(B\) be the set of all disjoint open subintervals of \(A\). Then

(a.) \(A\) and \(B\) are similar.
(b.) There is no onto map from \(A\) to \(B\), hence \(A\) and \(B\) are not similar.
(c.) There is no onto map from \(B\) to \(A\), hence they are not similar.
(d.) There is no one-to-one map from \(A\) to \(B\), hence they are not similar.

33. Let \(\mathbb{Z}\) denote the set of integers and \(\mathbb{Z}_{\geq 0}\) denote the set \(\{0,1,2,3, \ldots\}\). Consider the map \(f: \mathbb{Z}_{\geq 0} \times \mathbb{Z} \rightarrow \mathbb{Z}\) given by \(f(m, n)=2^{m} \cdot(2 n+1)\). Then the map \(f\) is

(a.) Onto (surjective) but not one-to-one (injective).
(b.) One-to-one (injective) but not onto (surjective).
(c.) Both one-to -one and onto.
(d.) Neither one-to-one nor onto.

34. Which of the following is necessarily true for a function \(f: X \rightarrow Y\)?

(a.) If \(f\) is injective, then there exists \(g: Y \rightarrow X\) such that \(f(g(y))=y\) for all \(y \in Y\).
(b.) If \(f\) is surjective, then there exists \(g: Y \rightarrow X\) such that \(f(g(y))=y\) for all \(y \in Y\).
(c.) If \(f\) is injective and \(Y\) is countable, then \(X\) is finite.
(d.) If \(f\) is surjective and \(X\) is uncountable, then \(Y\) is countably infinite.