Discrete Structures

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CARLETON UNIVERSITY SCHOOL OF COMPUTER SCIENCE WINTER 2013
COMP. 3803 - DISCRETE STRUCTURES: II ASSIGNMENT I DUE: FRIDAY JAN 25, 2013
__________________________________________________________________________________
Assignment Policy: Late assignments will not be accepted. You are expected to work on the assignments on your own. Past experience has shown conclusively that those who do not put adequate effort into the assignments do not learn the material and have a probability near 1 of doing poorly on the exams. Important note: When writing your solutions, you must follow the guidelines below.
• • • •

The answers should be concise, clear and neat. Make sure that your TA can read your solution. Please submit the solutions in the order of the problems, the solution to Problem 1, then to Problem 2 and so on. When presenting proofs, every step should be justified. Assignments should be stapled or placed in an unsealed envelope with your name and student number.

Substantial departures from the above guidelines will not be graded.

1. 2. 3. 4. 5.

Prove that the sum of n real rational numbers is rational if all of them are rational. Does the reverse hold true? What can you say about the rationality of the product of n rational numbers? Prove that if n is a positive integer, then n is odd if and only if 5n+6 is odd. Show by induction that n5 – n is divisible by 5 for all n ≥ 0. Show by induction that n2 – 1 is divisible by 8 whenever n is an odd positive integer. If possible, give examples of: a) A graph with 4 vertices whose degrees are 1, 2, 3 and 3. b) A graph with 6 vertices whose degrees are 2, 4, 3, 3, 4 and 5. If it is not possible, explain why. Let A = {0,1} and B be a countable set. Prove that the set of all functions from A to B is countable. Let A be the set of all even natural numbers, and B be the set of natural numbers divisible by 3. Prove…...

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