Let T be the set of all sequences {an} of elements of Z. Prove the follow- ing. (i) T is an integral domain with respect to addition and multiplication defined by for all {an}, {bn} ET, {an} + {bn} = {an} {bn} = {en}, {an + bn} . where cn = Cizo aibn-i. (ii) To = {{an} € T | ai = 0 for all but a finite number of indices} is a subring with identity. (iii) The element (1,1,0,...) is a unit in T, but not in To. (iv) (2, 3, 1, 0, 0,...) is irreducible in T, but not in To.

Using " Proof by contradiction" Prove that the multiplicative identity in a field is UNIQUE   Course: Abstract Algebra(Modern Algebra), Discrete Maths, Real Analysis

The circled one Addition of real numbers

proof by cases

Mathematical Induction

Challenge Problem. 1 Prove that if z is a real number such that z+ is an integer, then for all natural numbers n, 2" + is an integer. z"

Intro to Elementary Number Theory homework problem.    Topics covered prime numbers linear Diophantine equations systems of linear Diophantine equations.