MATH 521, Spring 2014, Assignment 2Due date: Wednesday, February 12Problems for submission:1. A real number is irrational if it cannot be written in the form a/b,a, b ∈ Z, b 6= 0. The set of irrational numbers is given by J = R \ Q.(a) Prove that, if p ∈ Q and q ∈ J then p + q ∈ J and, if p 6= 0,pq ∈ J.(b) Prove that there is an irrational number r ∈ J in between anytwo rational numbers p, q ∈ Q.(c) Prove that the irrational numbers are uncountable. (Hint: Showthat if A is an uncountable set and B is countably infinite thenthe set A \ B is uncountable.)(d) Do the irrational numbers form a field? Either prove the claimthat do form a field or derive a contradiction.2. Prove that if S is a countably infinite set then the power set P(S) isan uncountable set.3. Consider an ordered field F with additive identity 0 ∈ F and multi-plicative identity 1 ∈ F. We define the absolute value of x ∈ F tobe|x| =x, if x ≥ 0−x, if x < 0.Prove the following for any x, y ∈ F:(a) −x = (−1) · x.(b) x + x = 2 · x where 2 = (1 + 1) ∈ F.(c) |x| ≥ 0 with |x| = 0 if and only if x = 0.(d) |x + y| ≤ |x| + |y|(e) max(x, y) =12· (x + y + |x − y|) where (1/2) = 2−1(f) min(x, y) =12· (x + y − |x − y|) where (1/2) = 2−14. Suppose that X is an ordered set with the least-upper-bound propertyand that Sn⊆ X, n ∈ N, is a family of subsets which are boundedabove by a common upper bound. Consider the setS = {sup(Sn) | n ∈ N}.(a) Prove that sup(Sn) ≤ sup(S) for all n ∈ N.(b) Is it possible for the inequality sup(Sn) < sup(S) to be strictfor all n ∈ N (i.e. that no supremum of Snattains the globalsupremum of S)? Justify your answer.5. Suppose S ⊆ X where X is an ordered set. Either give an exampleof an S satisfying the given properties or prove that no such set mayexist:(a) S is bounded from above and below, and yet neither max(S) normin(S) exist.(b) min(S) and max(S) exist and max(S) = min(S).(c) min(S), max(S), inf(S), and sup(S) exist and inf(S) < min(S) <max(S) < sup(S).6. Chapter #1, Question #5 in Rudin.7. Chapter #1, Question #8 in Rudin.Honors Question Chapter #1, Question #9 in