Assignment 9 for COT3100

Assignment 9 for COT3100

Unit 1: Logic and Sets

1. (10) Draw a truth table for the compound statement .
2. (15) Use the laws of logic to show that .
3. (15) Show by any valid method except Venn diagram that .
4. (35) Given the sets and :
   • (5) Determine which one is a subset of the other.
   • (20) Prove that subset property by universal generalization.
   • (10) Prove by counter-example that they are not equal sets.
5. (25) Accepting the premises:

Conclude .

Unit 2: Numbers

6. (25) Prove by induction that for positive integers , .
7. (15) Prove that if 7 divides with remainder 1, 7 divides  with remainder 3.
8. (20) Taking into account identical letters, how many ways are there to arrange the word PARVAMONSTRA that both begin and end with consonants?
9. (15) How many unique combinations of monsters can a small monster collector capture, if that collector:
   • Has 22 small monster containment devices
   • Intends to use all of those devices
   • Has access to Earth, Fire, Ice, and Steam type small monsters
   • Intends to capture at least three Ice, at least two Earth and at most two Steam type small monsters

For questions 11 and 12, a small monster collector has captured thirty-one Anachronism-type small monsters.  Each Anachronism-type small monster has a 27% chance of being a Phlogiston-subtype and a 47% chance of being an Aether-subtype; it cannot be both.

10. (5) What is the probability that exactly seven of the captured small monsters are Phlogiston-subtypes?
11. (10) What is the probability of all but five of the captured small monsters being either Phlogiston- or Aether-subtypes, with those three being plain Anachronism-type monsters of neither subtype?
12. (10) What is the probability that all thirty-one captured small monsters are either Phlogiston- or Aether- subtypes, with no plain Anachronism-type monsters captured?

Unit 3: Relations and Functions

13. (15) Let with .  Characterize R in terms of whether it is reflexive, irreflexive, symmetric, anti-symmetric, transitive, complete, any sort of ing relation, and/or an equivalence relation.  This is not a formal proof, but briefly explain your reasoning.
14. (15) Let with , that is, x and y round up to the sane number.  Characterize R in terms of whether it is reflexive, irreflexive, symmetric, anti-symmetric, transitive, complete, any sort of ing relation, and/or an equivalence relation.  This is not a formal proof, but briefly explain your reasoning.
15. (15) Let with ; that is,  returns the square of  rounded up.  Characterize  in terms of whether it is injective, surjective and/or bijective.  This is not a formal proof, but briefly explain your reasoning.