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 functio...
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