Countable and Uncountable

Here we collect the various examples given in books and options asked in CSIR, GATE, TRB, JAM examinations about countable sets. 

Find which of the following sets are countable? 

1. The set of rational numbers $\mathbb{Q}$ 

2. The set of natural numbers $\mathbb{N}$

3. The set of real numbers $\mathbb{R}$

4. The set of prime numbers $\mathbb{P}$

5. The set of complex numbers with unit modulus.

6. The set of algebraic numbers.

7. The set of transcendental numbers.

8. The set of all polynomials with integer coefficients.

9. The set of all polynomials with rational coefficients.

10. The set of all polynomials over $\mathbb{R}$ with rational roots.

11. The set of all monic polynomials over $\mathbb{R}$ with rational roots.

12. The product set $\mathbb{N} \times \mathbb{N}$

13. The product set $\mathbb{N} \times \mathbb{R}$

14. $\wp(\mathbb{N})$, The set of all subsets of $\mathbb{N}$ 

15. The set of all finite subsets of $\mathbb{N}$ 

16. The set of all functions from   $\mathbb{N}$ to $\mathbb{N}$ 

17. The Set of all sequences with zeros and ones

18. The cantor set.

19. Let $X$ denote the two-point set $\{0,1\}$ and write $X_{j}=\{0,1\}$ for every $j=1,2,...$, Let $Y=\left(\prod_{j=1}^{\infty} X_{j}\right)$. Which of the following is/are true? 

(a) $Y$ is countable 

(b) card $Y=$ Card $[0,1]$

(c) $\bigcup_{n=1}^{\infty}\left(\prod_{j=1}^{n} X_{j}\right)$ is uncountable 

(d) $Y$ is uncountable

20. Which of the following subsets are uncountable? 

(a) $\{(a,b) \in \mathbb{R}^2| a \le b\}$ 

(b) $\{(a,b) \in \mathbb{R}^2| a + b \in \mathbb{Q}\}$ 

(c) $\{(a,b) \in \mathbb{R}^2|a b \in \mathbb{Z}\}$ 

(d) $\{(a,b) \in \mathbb{R}^2| a, b \in \mathbb{Q}\}$

 21. Which of the following sets of functions are uncountable? 

(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) $\{f \mid f: \mathbb{N} \rightarrow\{1,2\}, f(1) \leq f(2)\}$

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

(a) $P(A)=\phi$ for some $A$. 

(b) $P(A)$ is a finite set for some $A$. 

(c) $P(A)$ is countable set for some $A$ 

(d) $P(A)$ is uncountable set for some $A$.

23.  Which of the following subsets are uncountable? 

(a) $\{x\in \mathbb{R}|\log (x) =\frac{p}{q} \, for\, some \, p, q \in \mathbb{N}\}$ 

(b) $\{x\in \mathbb{R}|\sin^nx+\cos^nx=1 \, for\, some \, n \in \mathbb{N}\}$

(c) $\{x\in \mathbb{R}|x=\log (\frac{p}{q}) \, for\, some \, p, q \in \mathbb{N}\}$

(d) $\{x\in \mathbb{R}|x=\cos (\frac{p}{q}) \, for\, some \, p, q \in \mathbb{N}\}$