A. Given the existence of a set, R, which is the set of real numbers, with the operations of addition and multiplication, and the orderrelation < onRsuch that, for all x, y, zR,there are the following axioms:

Axiom 1: (x + y) + z = x + (y + z);(x * y) * z = x * (y * z)

Axiom 2: x + y = y + x; x * y = y * x

Axiom 3: x * (y + z)=(x * y)+(x * z)

Axiom 4: There is a unique element 0 R such that 0 + x = x for all x R.

Axiom 5: For each x R, there is a unique y R such that x + y = 0, and we write y = -x.

Axiom 6: There is a unique element 1 R such that x * 1 = x for all x R and 0 1.

Axiom 7: For each x R, with x 0, there is a unique element y R such that x * y = 1, and we write y = 1/x.

Axiom 8: x < y implies x + z < y + z

Axiom 9: x < y and y < z implies x < z

Axiom 10: For x, y R, exactly one of the following is true: x < y, y < x, or x = y.

Axiom 11: x < y and z > 0 implies xz < yz

1. Let aand cbe real numbers, with a. Using the axioms of the real number system given in part A, prove there exists a real number, b,so that a.

B. Create two subsets of the real numbers, C and D, where C is unbounded and does not contain all the real numbers and D is bounded and infinite.

1. State the supremum and infimum ofboth C and D.

2. State the supremum and infimum of CU D, or explain why they do not exist.

3. State the supremum and infimum of CD, or explain why they do not exist.

C. Prove the following claim, using proof by induction. Show your work.

Let d = 22. If a = 2d+ 1 and b = d + 1, then aⁿ - b is divisible by d for all natural numbers n.