Homework 5 with Problems - Discrete Structures | CMSC 250, Assignments of Discrete Structures and Graph Theory

Download Homework 5 with Problems - Discrete Structures | CMSC 250 and more Assignments Discrete Structures and Graph Theory in PDF only on Docsity! Fall 2009 CMSC 250: Homework 5 Perlis Due at the start of discussion section Wednesday, October 7, 2009. Please staple your homeworks together if you use more than one page. Problem 1. (1 point) Print your name , section number , and your TA’s name . Problem 2. (1 point) When is the first exam? | Problem 3. (30 points) For each of the following statements either (1) write TRUE and explain why, or (2) write FALSE and give a counterexample. a. For any two real numbers 𝑟 and 𝑠, if 𝑟 < 𝑠 there is a real number 𝑥 such that 𝑟 < 𝑥 < 𝑠. b. For any real number 𝑥, 𝑥 − 𝑥 < 1. c. For any positive real number 𝑥, 𝑥 𝑥 > 0. d. For any real numbers 𝑥 and 𝑦, if 𝑥 − 𝑦 = 0, then 𝑥 − 𝑦 = 0. e. For any real numbers 𝑥 and 𝑦, if 𝑥 𝑦 = 0, 𝑥 > 0 and 𝑦 > 0, then 𝑥 < 𝑦. f. For any integer 𝑎 and real number 𝑥, if 𝑎 divides 𝑥 , then 𝑎 divides 𝑎𝑥 . Problem 4. (20 points) Prove that for any integers 𝑚 > 1 and 𝑛 > 1, 𝑚𝑛 − 1 is not divisible by either 𝑚 or 𝑛. Problem 5. (24 points) a. Prove 5|𝑛2 → 5|𝑛. Use the unique factorization theorem. b. How would you modify your proof in (a) to prove 17|𝑛2 → 17|𝑛. (Describe the modifications, but do not actually write out the proof.) Problem 6. (24 points) a. Prove 5|𝑛2 → 5|𝑛. Do not use the unique factorization theorem. b. How would you modify your proof in (a) to prove 17|𝑛2 → 17|𝑛. (Describe the modifications, but do not actually write out the proof.)