Math 521, Spring 2014, Term Test I (Sample)Analysis IDate: Friday, February 21Name (printed):UW Student ID Number:Instructions1. Fill out this cover page.2. Answer questions in the space pro-vided, using backs of pages for over-flow and rough work.3. Show all work required to obtainyour answers.4. Unless otherwise stated, you mayuse any theorem or result derived inclass.FOR EXAMINERS’ USE ONLYPage Mark2 /∞3 /∞4 /∞5 /∞Total /∞Math 521, Term Test I, Spring 2014 Page 2 of 5 Name:1. Definitions and Theorems:(a) Let P and Q be statements which have a truth value and suppose that P =⇒ Q.State the contrapositive.(b) Let P and Q be statements which have a truth value and suppose that P =⇒ Q.State the converse.(c) Define what it means for a set S to be countably infinite.(d) Define what it means for a set S to be uncountable.(e) Define what it means for S to be bounded from below, where S ⊆ X is nonemptyand X is an ordered set.(f) Suppose X is an ordered set and S ⊆ X is nonempty and bounded above. Definesup(S).(g) Suppose X is an ordered set and S ⊆ X is nonempty and bounded below. Defineinf(S).(h) Define what it means for an ordered set X to have the least-upper-bound property.(i) Define what it means for a set X to be ordered.(j) State the multiplication axioms for a set F to be a field.(k) State the Cauchy-Schwarz Inequality.2. True/False:(a) The contrapositive is never a valid proof method. [True / False](b) The set of rational numbers Q does not have the least-upper-bound property [True/ False](c) Every infinite subset of a countable infinite set is countable infinite. [True / False](d) The power set P(S) of a countably infinite set S is countably infinite. [True /False](e) In any field F, the additive identity 0 ∈ F is unique. [True / False](f) Let F be a field. Then, for any element x ∈ F, x 6= 0, the multiplicative inversex−1is unique. [True / False](g) In any ordered field F, x > 0 and y > z implies that x ·y > x · z. [True / False]3. Set Proofs (Note: Venn diagrams are a helpful aid but do not constitute a proof!)Let A, B, and C be sets.(a) Prove that A ⊆ B =⇒ A \ B = ∅.(b) Prove that A ⊆ Bc=⇒ A ∩B = ∅(c) Prove that (A ∪ B)c= Ac∩ Bc.(d) Prove that (A \ B) ∪ (A \ C) = A \(B ∩ C)(e) Prove that (A ∪ B) \ (A ∩ B) = (A \ B) ∪ (B \ A).(f) Prove that A ∪ B ⊆ A ∩ B =⇒ A = B.2Math 521, Term Test I, Spring 2014 Page 3 of 5 Name:4. Density(a) Use the Archimedean property of the reals to prove that, for any p, q ∈ R satisfyingp < q, there is an r ∈ Q such that p < r < q.(b) Suppose that p, q ∈ R \ Q and p < q. Prove that there is an r ∈ R \ Q such thatp < r < q. (Note: It is not sufficient to consider the midpoint, e.g. p = −√2 andq =√2 butp + q2= 0 is rational.)5. Countability(a) Prove that, if A is a countably infinite set and B ⊆ A is infinite, then B is countablyinfinite.(b) Prove that, if A is an uncountable set and B ⊆ A is countably infinite, then A \Bis an uncountable set.(c) Let I denote the set of all intervals [a, b] where a, b ∈ Q and b > a. Is I countableor uncountable? Prove your claim.(d) Let S denote the set of all potentially non-terminal decimal numbers between 0 and1 which can consist of only the numbers 1, 4, and 7 (e.g. s = 0.7117417471... ∈ S).Is S countable or uncountable? Prove your claim.6. Ordered Sets(a) Suppose that X is an ordered set and every nonempty S ⊆ X which is boundedbelow has the property that inf(S) ∈ X. Prove that X has the least-upper-boundproperty.(b) Suppose X is an ordered set and S ⊆ X is nonempty and bounded above. Provethat sup(S) is unique.(c) Suppose X is an ordered set with the least-upper-bound property and A, B ⊆ Xare nonempty. Suppose that x < y for every x ∈ A and y ∈ B. Prove thatsup(A) ≤ inf(B). Is it possible for equality to be attained?(d) Suppose X is an ordered set with the least-upper-bound property and {Sn}n∈Nis a family of subsets which are bounded below by a common bound. DefineS = {inf(Sn) | n ∈ N}. Prove that inf(S) exists and inf(S) ≤ x for all x ∈[n∈NSn.(e) Define S = {(x1, x2) ∈ R2|px21+ x22≤ 1} and define the following ordering onR2: For x = (x1, x2) and y = (y1, y2) we have x > y if x1> y1or if x1= y1andx2> y2. (i) Determine sup(S) and inf(S). (ii) If we instead considered the setS0= {(x1, x2) ∈ R2|px21+ x22< 1}, do sup(S0) and inf(S0) still exist?(f) Suppose S ⊆ R is nonempty and bounded below. For a fixed λ < 0, defineEλ= {λx | x ∈ S}. Prove that sup(Eλ) = λ inf(S).3Math 521, Term Test I, Spring 2014 Page 4 of 5 Name:7. FieldsLet F denote a field. Suppose x, y, z ∈ F. Using only the field axioms, prove thefollowing:(a) 1 ∈ F is unique (i.e. x · y = x implies y = 1)(b) (x−1)−1= x(c) x + y = x + z implies y = z(d) 0 · x = 0.(e) 0 6= 1(f) (x + y)2= x2+ 2 · x ·y + y2, where 2 = 1 + 1 and x2= x · x(g) (x + y) · (x −y) = x2− y2, where x2= x · x and x −y = x + (−y)8. Miscellaneous(a) Prove that√6 is irrational.(b) How many sets are there in P(P({1}))? List them all. (Note: {1} is the singletonset with only the element ‘1’.)(c) Consider the family of sets {Sα}α∈Awhere Sα= {x ∈ R | α ≤ x < 1 + α} andA = {α ∈ R | 0 < α < 1}. Determine[α∈ASαand\α∈ASα.(d) Consider the family of sets {Sα}α∈Awhere Sα= {x ∈ Q |√2 − α < x <√2 + α}and A = {α ∈ R | 0 < α < 1}. Determine[α∈ASαand\α∈ASα.4Math 521, Term Test I, Spring 2014 Page 5 of 5 Name:THIS PAGE IS FOR ROUGH