11. Which set is countable?

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:

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:

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:

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:

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.

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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:

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 \(\mathbb{C}\).

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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:

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:

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