- Rouche's theorem
- Practice Problems-1 (PG Mathematics)
- Practice problems -2(PG Mathematics)
- Practice problems -3(PG Mathematics)
- Problems on Field Extensions and Galois theory
- Some CSIR problems
- Some Problems with Hints...
- NBHM Problem...
- Books(Authors) list for TRB Exams
- Uniform Convergence: Examples with graphs
- Practice problems - 4(PG mathematics)
- Leibniz rule: Differentiation under the integral sign
- Integration-Summation
- Primitive roots of unity
- Euler's theorem on Number Theory
- Sylow's theorem: Number of sylow subgroups
- Countable and Uncountable
- Gershgorin circle theorem: A bound on eigenvalues
- Descriptive type questions on Mathematics competitive examinations
- Stirling's approximation
- Thomae's function
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 $\
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