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Homework: MATH 201 Homework 0: Due: Thursday, September 1st Briefly relate (in one or two paragraphs) information about yourself that will help me get to know you. If you wish, you may let the following questions serve as a guide: When did you take Math 161 and 162 (or their equivalents)?; why are you taking Math 201 now? (for example: "major requirement", “minor requirement”, "just for fun because I love mathematics", "nothing else fits my schedule", "my parents forced me to take this course", "I am looking for an easy A to raise my gpa"); what is your major?; what is your career goal?; what has been the nature of your previous experience with math either in high school or in college (that is, have you enjoyed math in the past?). (Please post your response as a private message in Piazza no later than midnight, Wednesday. For “Subject” write “201 Homework 0”) Homework 1: Due: Thursday, 8th September Read section 2.3 (pp 35 – 45) and section 2.8 (optional) of Belcastro. In solving the homework exercises, be certain to follow the proper procedures discussed in class as well as in the document How to Write Good Proofs. 1. Let p represent the proposition "Henry VIII had six wives". Let q represent the proposition "Stephen Colbert was born in Iceland". (a) Connect these two propositions with OR. Is the resulting compound proposition true or false? (b) Now connect them with AND. Is this compound proposition true or false? (c) Is the 'opposite' of p true or false? (d) Is p ⇒ q true? (e) Is q ⇒ p true? (f) Is (~p) ⇒ (~q) true? (g) Is (P ∧ (P⇒Q)) ⇒ Q true? 2. Prove the following (adhering to the rules of good proofs). Let a and b be non-negative real numbers. Then a = b if & only if √𝑎𝑏 = 𝑎+𝑏 . 2 (Hint: Modify the inequality proof that we discussed in class.) 3. Construct a truth table for the following compound proposition: ¬r∧¬p)∧(p∨r) (¬p∨q)∧(q → 4. Give a precise and clear proof of the Pythagorean theorem using Bhāskarāchārya’s clever geometric idea: Behold! Bhāskarāchārya (12th century Indian mathematician and astronomer) Homework 2: Due: Thursday, 15th September Read section 2.5 (pp 46 – 48) of Belcastro. In solving the homework exercises, be certain to follow the proper procedures discussed in class as well as in the document How to Write Good Proofs. 1. Using the method of proof by contradiction show that, if a is rational and b is irrational, then a + b is irrational. 2. Suppose a, b ∈ Z. If (a2 + b2) is a multiple of 4, then a and b are not both odd. Use the method of proof by contradiction. 3. Suppose a ∈ Z. If a2 − 2a + 7 is even. Prove, using the method of contradiction that a is odd. 4. Prove that there is no rational solution to the equation x3 + x + 1 = 0. Use the method of proof by contradiction. Homework 3: Due: Thursday, 22nd September Prepare for Test I on Thursday. In solving the homework exercises, be certain to follow the proper procedures discussed in class as well as in the document How to Write Good Proofs. 1. Consider the statement. If I am a Martian and hungry, then I watch The Brain Dead or do my calculus homework. Write the inverse, converse and contrapositive of this statement. Use deMorgan’s laws when appropriate to simplify your proposition. 2. Proof without words: Using the clever picture below, give a precise and clear proof of the the Arithmetic Mean – Geometric Mean Inequality 𝒂+𝒃 𝟐 ≥ √𝒂𝒃 with equality iff a = b 3. Prove by cases that, if x is a real number such that < −1. 𝒙𝟐 −𝟏 𝒙+𝟐 > 𝟎, then either x > 1 or −2 < x 4. The following is a Bogus Well-Ordering Principle Proof The Fibonacci numbers 0, 1, 1, 2, 3, 5 ,8,13,… are defined as follows. Let F(n) be the nth Fibonacci number. Then F(0) = 0 F(1) = 1 F(n) = F(n−1) + F(n−2), for n ≥ 2 (*) Identify which step(s) contain the logical error! Explain! Bogus Claim: Every Fibonacci number is even. The proof is by the WOP. Let Even(n) mean that F(n) is even. Let C be the set of counterexamples to the assertion that Even(n) holds for all n ≥ 0. That is C ={ n ≥ 0 | ¬Even(n))} We prove by contradiction that C is empty. ; Assume that C is not empty. By the WOP, there is a least nonnegative integer, m∈C. ; Then m≥0, since F(0)=0 is an even number. Now, suppose m ≥ 2 so the definition (⋆) of F(m) applies. In this case, both F(m − 1) and F(m − 2) are both even, since m is the minimum counterexample such that F(m) is not even. But by (⋆) in the case that n=m, we see that F(m) is the sum of two even numbers, so it is also even; thus Even(m) is true. This deduction contradicts the condition in the definition of m that NOT(Even(m)) is true. This contradiction implies that C must be empty. Hence, F(n) is even for all n ≥ 0. Homework 4 Due: Thursday, 6th October 1. How many positive integers not exceeding 1000 are (a) divisible by 7? (b) divisible by 49? (c) divisible by 3 but not by 4? 2. Proof without words: Using the clever picture below, give a precise and clear proof of the well-known fact that e > e . 3. Prove that no integer in the sequence 11, 111, 1111, 11111, 111111, . . . is a perfect square. (Hint: 111 · · · 111 = 111 · · · 108+ 3 = 4k+ 3.) 4. A number is perfect if it is equal to the sum of its positive divisors, other than itself. For example, 6 is perfect, because 6 = 1 + 2 + 3. Similarly, 28 is perfect, because 28 = 1 + 2 + 4 + 7 + 14. Explain why n = 2k−1(2k − 1) is perfect if 2k − 1 is prime. [Hint: Begin by finding the divisors of n.] Homework 5 Due: Tuesday, 18th October 1. Proof without words. Watch the MAA animation of Siu’s proof without words of the sum of squares formula. Next, explain the proof in your own (precise and clear) words. 3. Using induction, prove the following: A total of n democrats and republicans stand in line. If the person at the head of the line is a democrat and the person at the end of the line is a republican, then somewhere in the line, a republican must stand next to a democrat. Homework 6 Due: Thursday, 10th November Study the MIT handout on paradoxes and infinity. Read section 15.3 of Ducks. 1. Read about the Euclidean algorithm and the extended Euclidean algorithm. (For example, at https://www.math.rutgers.edu/~greenfie/gs2004/euclid.html or https://www.youtube.com/watch?v=hB34-GSDT3k or https://www.youtube.com/watch?v=JUzYl1TYMcU ) Next, compute gcd(53, 77), and use your computation to find integers x and y such that gcd(53, 77) = 53x + 77y. 2. Prove: If A, B are finite sets of the same cardinality then any injection or surjection from A to B must be a bijection. 3. Let T: X→Y and S: Y→ Z be mappings and assume that 𝑆 ∘ 𝑇 is an injection. (a) Prove that T must be injective. (b) By means of a counter-example, show that S need not be injective. 4. We proved in class that N×N is countable. Construct an alternate proof of this fact by showing that the function ϕ: N×N → N defined as ϕ(m, n) = 2n−1 (2m −1) is bijective. Homework 7 Due: Thursday, 17th November Prepare for Test 3. 1. Explain the “proof without words” displayed to the left. 2. In the land of Oz it has been decreed that every password must be exactly 11 characters in length, contain exactly one capital letter (A, B, C..., Z), exactly one lower case letter (a, b, c, ..., z), and exactly one of three special characters (#, $, @). The remaining characters must be digits (0, 1, 2,... 9). How many such passwords exist? 3. [MIT 24.118 hw problem] Consider the following infinite tree: (When fully spelled out, the tree contains one row for each natural number. The zeroth row contains one node, the first row contains two nodes, the second row contains four nodes, and, in general, the nth row contains 2n nodes.) (a) Is there a 1-1 correspondence between the nodes of this tree and the natural numbers? (b) Is there a 1-1 correspondence between the branches of this tree and the natural numbers? (Don’t forget to justify your answers!) Note: A branch is an infinite sequence of nodes which starts at the top of the tree and contains a node at every row, with each node connected to its successor by an edge. (Branches can be represented as infinite sequences of zeroes and ones.) 4. Let X = set of all polynomials of degree ≤ 5 with real coefficients. Define the following two operations on X. For each of the following decide if X is closed under the given operation. Give proof or counter-example. (a) composition of functions (b) p ♠ q = p + q – pq + 4x + 1 (c) p ☼ q = p(2x)q(3x + 1) Georg Cantor, father of modern set theory “Todd, trust math. As in Matics, Math E. First-order predicate logic. Never fail you. Quantities and their relation. Rates of change. The vital statistics of God or equivalent. When all else fails. When the boulder's slid all the way back to the bottom. When the headless are blaming. When you do not know your way about. You can fall back and regroup around math. Whose truth is deductive truth? Independent of sense or emotionality. The syllogism. The identity. Modus Tollens. Transitivity. Heaven's theme song. The night light on life's dark wall, late at night. Heaven's recipe book. The hydrogen spiral. The methane, ammonia, H2O. Nucleic acids. A and G, T and C. The creeping inevitability. Caius is mortal. Math is not mortal. What it is is: listen: it's true.” – David Foster Wallace, Infinite Jest The seeming absence of any ascertained organizing principle in the distribution of the succession of the primes had bedeviled mathematicians for centuries and given Number Theory much of its fascination. Here was a great mystery indeed, worthy of the most exalted intelligence: since the primes are the building blocks of the integers and the integers the basis of our logical understanding of the cosmos, how is it possible that their form is not determined by law? Why isn't 'divine geometry' apparent in their case? - A. Doxiadis, from the novel Uncle Petros and Goldbach's Conjecture, p. 84 (Faber 2000) Course Home Page Department Home Page Loyola Home Page