COMP1805A (Summer 2018) “Discrete Structures I” Assignment 2 Please ensure that you include your name and student number on your submission.
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COMP1805A (Summer 2018) “Discrete Structures I” Assignment 2 Please ensure that you include your name and student number on your submission. Your submission must be created using Microsoft Word, Google Docs, or LaTeX. Your submission must be saved as a single “pdf” document and have the name “lastname.studentID.a2.pdf “ Do not compress your submission into a “zip” file. Late assignments will not be accepted and will receive a mark of 0. Submissions written by hand, compressed into an archive, or submitted in the wrong format (i.e., are not “pdf” documents) will receive a mark of 0. Always show all of your work. The due date for this assignment is May 29 th, 2018, by 11:30pm. 1. Determine whether or not the following arguments are valid. If they are valid, then state the rules of inference used to prove validity. If they are invalid, outline precisely why they are invalid. [12 marks] a. If it is summer, then it is humid outside. If Santa is delivering presents, then it is not the case that it is hot and humid outside. It is currently hot outside and Santa is delivering presents. Therefore, it must not be summer. b. If it is Thursday, then I do not have to go to class. I do not go to class. Therefore, it must be Thursday. c. Everyone taking COMP1805 loves math. Everyone taking STAT 2507 loves math. Therefore, everyone in STAT 2507 is taking COMP 1805. 2. Prove that √11 − 1 is an irrational number, using a proof by contradiction. Remember that the definition of a rational number is a number that can be written as a fraction, where the numerator and denominator are both integers, and the fraction is in lowest form. Use prime factorization. [17 marks] 3. Prove that √7 is an irrational number using a proof by contradiction and proof by cases for √7 (i.e.do not use prime factorization here). [30 marks] 4. Prove by indirect proof that when n is an integer, if n 3+3 is odd, then n is even. Show all your work (no calculators allowed). [5 Marks] COMP1805A (Summer 2018) “Discrete Structures I” Assignment 2 5. Find the error in the following proof that every positive integer equals the next largest positive integer: “Proof: Let P(n) be the proposition ‘n=n+1’. Assume that P(n) is true, so that n=n+1. Add 1 to both sides of this equation to obtain n+1=n+2. Since this is the statement P(n+1), it follows that P(n) is true for all positive integers n.” [1 mark] 6. For integer x, such that -2≤x≤2, prove that y<0, where y= x 2 –2x – 44. [6 marks] 7. Prove by induction that 1+3+5+…+(2n-1) = n 2 , for all positive integers n. [6 marks] 8. Suppose a,r∈ℝ, r≠0 and r≠1. Prove by induction that a+ar+ar2+…+arn-1= a (1 –
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