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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 $\
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NBHM Problem...

Let T  be a nilpotent linear operator on the vector space $\mathbb{ R}^5$  (i.e., $ T^k=0$  for some k). Let $d_i$  denote the dimension of the kernel of $ T^i$ . Which of the following can possibly occur as a value of $ (d_1,d_2,d_3)$ ?  1. (1,2,3) 2. (2,3,5) , 3. (2,2,4) , 4. (2,4,5) For the excellent proof using Jordan Canonical form visit  https://nbhmcsirgate.theindianmathematician.com/2020/04/nbhm-2020-part-c-question-26-solution.html#.XqCM3QR0R10.whatsapp Here we present a proof (incomplete in preciousness) on the fact "A linear transformation is completely determined by its behaviour on a basis."
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