Volume 55 | Number 3 | Year 2018 | Article Id. IJMTT-V55P525 | DOI : https://doi.org/10.14445/22315373/IJMTT-V55P525

Set theory is widely use in the mathematics to study sets. The number is grain of mathematics, which categories in the set of integers (Z), the set of natural number (positive integers) (N), the set of rational numbers (Q), and set of real number (R). Here is study of the natural number by using the arithmetic series distribution and by the set theory. So, study of number in sequential series and by set distribution is called as ‘number set theory’. Here we use the set theory is in slightly different way and in different notation. The theory study the arithmetic series problem, like Collatz conjecture / (3n+1) problem. Collatz conjecture is most famous (3n+1) arithmetic series problem. (3n+1) is one of the set which, we represent in number set theory is S{3n+1}. This set is divided into many sub-set for the deep understanding of the (3n+1) problem. Here is study of the collatz conjecture using the number set theory.

[1] Norman T. Hamilton and Joseph Landin, Set theory: the structure of Arithmetic, Allyn and Bacon, inc. Boston, 1961.

[2] Jeffrey C. Lagarias, Editor, The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010.

Joshi Ashok B, "Number Set Theory and Collatz Conjecture," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 55, no. 3, pp. 196-199, 2018. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V55P525