MATH 290 WRITING ASSIGNMENT 3 SOLUTIONS 1. Dene the Fibonacci sequence cfw_f1 , f2 , f3 , . . . by f1 = f2 = 1 and for n > 2, fn = n n fn1 + fn2 . Use induction to prove that for all natural numbers n, fn = , where 1+ 5 1 5 = and = . (Hint: and are the so
MATH 290 WRITING ASSIGNMENT 1 SOLUTIONS 1. Prove that for all real numbers a and b, |a + b| |a| + |b|. (Hint: This will be an example of a proof by cases.) Solution: Let a and b be real numbers. We must show that |a + b| |a| + |b|. In order to do this we
MATH 290 HOMEWORK 10 SOLUTIONS In this homework, we will prove from scratch (that is, without using LHopitals Rule) that lim n1/n = 1. 1. Prove that lim 2 = 0. Do this using the denition of convergence, and by invoking n1 the Archimedean Principle of the
MATH 290 HOMEWORK 9 SOLUTIONS Let A and B be sets, and suppose that f : A B . 1. Prove that for every X A, X f 1 (f (X ). Solution: Let X A and suppose that x X . If we let y = f (x), then by denition, y f (X ) so that f (x) f (X ). This implies that ther
MATH 290 HOMEWORK 8 DUE 22 APRIL 2010 Answer all of the following questions clearly and completely. You will be graded on both the answers themselves and on presentation of the answers. This includes style, grammar and spelling. There is to be no collabor
MATH 290 HOMEWORK 7 SOLUTIONS The following problems are based on Exercise 24 of Section 3.4, pp. 168-169 in the book. Let A be a set with partial order R. For each a A, let Sa = cfw_x A: xRa. Let F = cfw_Sa : a A. Then F is a collection of subsets of A,
MATH 290 HOMEWORK 6 SOLUTIONS 1. Dene the relation R on the integers Z by xRy if and only if either x|y or y |x. (a) Prove that the relation R is not transitive. (Hint: To show it is not transitive, you must nd three integers x, y , and z such that xRy an
MATH 290 HOMEWORK 5 SOLUTIONS Denition. An ordered triple of natural numbers (a, b, c) is a Pythagorean triple if a < b < c and a2 + b2 = c2 . The set of all Pythagorean triples is denoted by P . The numbers a and b are called the legs of the triple and c
MATH 290 HOMEWORK 4 SOLUTIONS 1. For each natural number n, let Mn = cfw_k Z: n|k . (In other words, for each natu
ral number n, Mn = cfw_. . . , 3n, 2n, n, 0, n, 2n, 3n, . . ..) Prove that
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Mn = cfw_0. (Hint: It is not sucient to just
MATH 290 HOMEWORK 3 SOLUTIONS 1. Prove that p is prime if and only if for all natural numbers a and b, either p divides a or p divides b whenever p divides ab. (Hint: You may use the denition of a prime number, the facts given in Exercises 9(a) and 9(b) o
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MATH 290 WRITING ASSIGNMENT 2 SOLUTIONS 1. Let cfw_a1 , a2 , a3 , . . . be a sequence of real numbers. Use induction to prove that for all natural numbers n, |a1 + a2 + a3 + + an | |a1 | + |a2 | + |a3 | + + |an |. Solution: The proof will be by induction.
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MATH 290 HOMEWORK 2 SOLUTIONS 1. Prove that for any integers a and b, a + b is even if and only if either both a and b are even or both a and b are odd. Solution: Let a and b be integers. (=) We will prove this direction by contrapositive. Suppose that it
MATH 290 HOMEWORK 1 SOLUTIONS 1. Show the following using a truth table. (a) (P Q) = R is equivalent to (P = R) (Q = R) (b) P = (Q R) is equivalent to (P Q) = R). Solution: (a). P T T T T F F F F Q T T F F T T F F (b). P T T T T F F F F Q T T F F T T F F
MATH 290 FINAL EXAM SOLUTIONS 1. (12 pts.) Prove that for all integers a, and b, and all nonzero integers c, ac|bc if and only if a|b. Which of the implications remains true if c = 0? Solution: Let a and b be integers and let c be a nonzero integer. (=) S
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Nicola Hodges David Walnut MATH 290-002 December 14th, 2010 Writing As ig e 10 s nm nt 1. Le A and B befinites ts with A~ Suppos f: A B t e B. e a. If f is one -to-one s , how that f is onto B. Let A and B be two finite Sets and A B. Assume f is a functio
Nicola Hodges David Walnut MATH 290-002 November 29, 2010 Writing Assignment 8 1. Prove that if xn L and y n M and r R, then x n y n L M Lemma 1: Suppose xnL. Then xn is bounded, hence there exists a Z such that for all n N, |xn| Z. Let e=1. Then for some
Nicola Hodges David Walnut MATH 290-002 November 21, 2010 Writing Assignment 8 1. Show that the relation R on the natural numbers given by aRb iff b=2ka for some integer k 0 is a partial ordering. Show that the relation on the natural numbers aRb, where b
Nicola Hodges David Walnut MATH 290-002 October 23rd, 2010 Writing Assignment 6 1. L et a and b be natural numbers, and let d be the smallest n atural number such that there exists integers x and y such t hat ax+by = d. Prove that d = GCD (a,b). ( Hint: L
Nicola Hodges David Walnut MATH 290-002 October 28, 2010 Writing Assignme n t 5
1. U se a form of induction to prove that for all natural numbers n 2 , eithe p rime or n has a prime factor 2. p with p Proof by Complete Induction. Base Case. If n=2, then c