numbers 2. Show that if a and b are positive integers, then there is a smallest positive integer of the form a-bk, k = Z. -Prove that both the sum and th

1. For any real number r and any integer r #k, show that a) ()=(¹), and b) () = (-1)k (r+k-1). Hint: use the definition of () from Page 137. r-k

In “Elementary Number Theory & Its Applications” by Kenneth Rosen, in chapter 1.1, the following is defined: The greatest integer in a real number x, denoted [x] is the largest integer function less than or equal to x. That is, [x] is an integer satisfying: [x] ≤ x

a) It is known that Q ͠ N (Rational Numbers have the same power as the Natural Numbers set N.).  So show that QxQ  ͠  N. ( QxQ have the same power as the N ) (Definition:  Let A and B be two sets.If there is a one-to-one function from A to B and at least one overlying function, it is said that A set has the same power as set B. and shown to be A ͠  B)   (It's abstract mathematics, please can you write step-by-step solutions I have a shortcoming in this regard.)

A number r ∈ R  is called algebraic provided for some n ∈ N there exists numbers  a0, a1, … , an ∈ Z such that r satisfies p( r ) = 0 where p ( x ) = a0 + a1x + a2x2 + ⋯ + anxn.   A number r ∈ R is called transcendental provided it is NOT algebraic prove the set of ALL algebraic numbers is countable and the set of ALL transcendental numbers is uncountable.

Question 6 Prove that a non-empty finite set S of real numbers has an infimum and a supremum and they are elements of S.