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

11. Which set is countable?

(a.) The set of all polynomials with real coefficients
(b.) The set of all subsets of a countably infinite set
(c.) The set \(A-B\) where \(A\) is uncountable but \(B\) is countable
(d.) The set of all finite subsets of \(\mathbb{N}\)

Explanation for Question 11:
a. \(R[x]\) is similar to \(R\), and both are uncountable. b. The set of all infinite subsets of \(\mathbb{N}\) is similar to the set of all sequences over \(\mathbb{N}\) and is uncountable. c. Take \(A = \mathbb{R}\) and \(B = \{0\}\). d. Let \(A_i = \{X \subseteq \mathbb{N} \ | \ |X| = i\}\). Note that \(A_i\) is countable \(\forall i\), and the set of all finite subsets of \(\mathbb{N} = \bigcup_{i=0}^{\infty} A_i\), which is a countable union of countable sets, is countable.

12. If \(F\) is the set of all functions defined on \(I_{n}=\{1,2,3, \ldots ., n\}\) where \(n \in \mathbb{N}\) with the range \(B \subseteq \mathbb{I}^{+}\) (the set of positive integers), then:

(a.) \(F\) is countable.
(b.) \(F\) is uncountable.
(c.) \(F\) is infinite.
(d.) \(F\) is countable if \(B\) is finite.

Explanation for Question 12:
Let \[ \begin{aligned} & A_n = \left\{f \ | \ f:\{1,2, \ldots, n\} \longrightarrow B \subseteq \mathbb{N}^{+}\right\} \\ \text{then} \ & \left|A_n\right| \leq \aleph_0^n = \aleph_0 \quad \forall n \geq 1 \text { as } |B| \leq \aleph_0 \\ F = & \bigcup_{n=1}^{\infty} A_n \end{aligned} \] Countable union of countable is countable \(\Rightarrow F\) is countable.

13. Select the correct statements:

(a.) Every countable set is similar to \(\mathbb{N}\).
(b.) The set of all disjoint open intervals is not similar to the set of real numbers.
(c.) The power set of \(\mathbb{N}\) is similar to the set of real numbers.
(d.) All of these statements are correct.

Explanation for Question 13:
\(|P(\mathbb{N})| = |R| = c\).

14. Let \(P_{n, m}\) be the set of polynomials of degree \(n\) with integral coefficients such that \(|a_{0}|+|a_{1}|+\ldots .+|a_{n}|=m\), where \(a_{i}\)'s are coefficients. Then:

(a.) \(P_{m, n}\) is countably infinite.
(b.) \(P_{m, n}\) is finite for some \(m, n\) only.
(c.) \(P=\bigcup\{P_{m n}: m, n \in \mathbb{N}\}\) is the set of all algebraic numbers.
(d.) \(P_{m n}\) is finite for all \(m, n \in \mathbb{N}\).

Explanation for Question 14:
Take \(m = n = 1\) and discard a and b. As \(P_{m,n}\) is a set of polynomials and not a set of algebraic numbers, c is also incorrect.

15. Let \(S=\{a \in \mathbb{R}: a\) is a recurring decimal number\} and \(T=\{b \in \mathbb{R}: b=\sqrt{\frac{p}{q}}\) for some distinct primes \(p\) and \(q\}\). Then:

(a.) \(S\) is countable but \(T\) is not.
(b.) \(T\) is countable but \(S\) is not.
(c.) Both \(S\) and \(T\) are countable sets.
(d.) Both \(S\) and \(T\) are uncountable sets.

Explanation for Question 15:
Irrational numbers do not have a recurring decimal, thus \(S\) contains only rationals, hence countable. Define \(f: T \rightarrow \mathbb{N} \times \mathbb{N}\) s.t. \(f(\sqrt{\frac{p}{q}}) = (p, q)\). Then \(f\) is \(1-1 \Rightarrow |T| \leqslant |\mathbb{N} \times \mathbb{N}| = \aleph_0\). \(\Rightarrow T\) is countable.

16. Consider the following statements:

1. Every infinite set is equivalent to at least one of its proper subsets.
2. If a set is equivalent to one of its proper subsets then it is an infinite set.

Code:

(a.) 1 is correct and 2 is incorrect
(b.) 2 is correct and 1 is incorrect
(c.) Both are correct
(d.) Both incorrect

Explanation for Question 16:
1. For any infinite set \(A\) and \(a \in A\), \(|A| = |A - \{a\}|\), hence they are similar. 2. Let \(X\) be a finite subset \(\Rightarrow |X| = n\) then for any \(Y \subset X, \ |Y| < n\). Hence there cannot be a bijection between them. Thus, they are non-similar.

17. Let \(A_{1}, A_{2}, \ldots, A_{n}\) be sets, where \(n\) is a fixed natural number. Consider the following statements:

(a.) If \(\displaystyle A=\bigcap_{i=1}^{n} A_{i}\) is countably infinite, then there exists at least one \(A_{i}\) for \(i=1,2, \ldots, n\) that is countable.
(b.) If \(\displaystyle A=A_{1} \times A_{2} \times \ldots \times A_{n}\) is countably infinite, then each \(A_{i}\) for \(i=1,2, \ldots, n\) is countable.
(c.) Both statements are correct.
(d.) Neither statement 1 nor statement 2 is correct.

Explanation for Question 17:
Let \(A_1 = \{(a, b) \ | \ a \in \mathbb{Q}, b \in \mathbb{R}\}\) and \(A_2 = \{(a, b) \ | \ a \in \mathbb{R}, b \in \mathbb{Q}\}\). \(A_1\) \& \(A_2\) are both uncountable, but \(A_1 \cap A_2 = \{(a, b) \mid a \in \mathbb{Q}, b \in \mathbb{Q}\}\) is countable. \(\Rightarrow\) a is false. If \(A_j\) for some \(j\) is uncountable, then \(A = A_1 \times \ldots \times A_n\) s.t. \(j \leqslant n\), is uncountable \(\Rightarrow 2\) is true.

18. How many statements is/are false?

1. Cardinality of \([0,1] \times [0,1]\) is the same as the cardinality of \(\mathbb{R}\).
2. Cardinality of \(\mathbb{R}\) is the same as the cardinality of irrationals.
3. Cardinality of \(\mathbb{R}\) is the same as the cardinality of \(\mathbb{C}\).

Code:

(a.) Zero
(b.) One
(c.) Two
(d.) Three

Explanation for Question 18:
All are true statements.

19. Let \(A\) be an uncountable subset of \(\mathbb{R}\) and \(B\) be a proper infinite subset of \(\mathbb{N}\). Define \(f: B \rightarrow A\) such that \(f\) is one-to-one. Then:

(a.) \(A\) and \(A-B\) are similar.
(b.) \(A\) and \(A-f(B)\) are similar.
(c.) \(\mathbb{R}\) and \(\mathbb{R}-f(B)\) are similar.
(d.) All of the above statements are correct.

Explanation for Question 19:
Note that for a one-one function, domain and range are similar. That is, \(f(B)\) is countable as \(B\) is countable. Implies that c is true. Also, for any uncountable set - a countable set is uncountable implies a, b, and d are true.

20. Let \(P_{n}\) be the set of all polynomials of degree \(n\) with integral coefficients. Then:

(a.) \(P_{n}\) is a finite set with \(n^{n}\) elements.
(b.) \(P_{n}\) is a finite set.
(c.) \(P_{n}\) is an infinite set if \(n>1\).
(d.) \(P_{n}\) is an infinite set for all \(n \in \mathbb{N}\).

Explanation for Question 20:
c and d are correct as the set of all constant polynomials is \(\mathbb{Z}\).