Discrete Mathematics I Tutorial 01 - Answer
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1 Discrete Mathematics I Tutorial 01 - Answer Refer to Chapter Let p and q be the propositions p : It is below freezing. q : It is snowing Write these propositions using p and q and logical connectives. a) It is below freezing and snowing. b) It is below freezing but not snowing. c) It is not below freezing and it is not snowing. d) It is either snowing or below freezing (or both). e) If it is below freezing, it is also snowing. f) It is either below freezing or it is snowing, but it is not snowing if it is below freezing. g) That it is below freezing is necessary and sufficient for it to be snowing. a) p q b) p q c) p q d) p q e) p q f) (p q) (p q) or (p q) g) p q Discrete Mathematics I Tutorial 1 1
2 2. Let p, q, and r be the propositions p : You have the flu. q : You miss the final examination. r : You pass the course. Express each of these propositions as an English sentence. a) p q c) q r e) (p r) (q r) b) q r d) p q r f) (p q) ( q r) a) If you have flu, you will miss the final examination b) You do not miss the final examination if and only if you pass the course c) If you miss the final examination, you will not pass the course d) You have flu, or miss the final examination, or also pass the course e) If you have the flue, you will not pass the course or if you miss the final examination, you will also not pass the course f) You have the flu and you miss the final examination, or either you do not miss the final examination and pass the course. 3. Determine whether these biconditionals are true or false. a) = 4 if and only if = 2. b) = 2 if and only if = 4. c) = 3 if and only if monkeys can fly. d) 0 > 1 if and only if 2 > 1. a) TRUE b) FALSE c) TRUE d) FALSE Discrete Mathematics I Tutorial 1 2
3 4. Add blanks to the suitable position to make the proposition more readable: a) p q s r q b) p q s r q q a) p q s r q p q s r ( q) p (q s) r ( q) p ((q s) r) ( q) p (((q s) r) ( q)) b) p q s r q q p q ( s) r q q (p q) ( s) r q q (p q) ( s) r (q q) ((p q) ( s)) r (q q) (((p q) ( s)) r) (q q) 5. For each of these sentences, determine whether an inclusive or or an exclusive or is intended. Explain your answer. a) Coffee or tea comes with dinner. b) A password must have at least three digits or be at least eight characters long. c) The prerequisite for the course is a course in number theory or a course in cryptography. d) You can pay using U.S. dollars or euros. e) Experience with C++ or Java is required. a) exclusive b) inclusive c) inclusive d) exclusive e) inclusive Discrete Mathematics I Tutorial 1 3
4 6. Construct a truth table for each of these compound propositions. a) p (q r) b) p p c) p q p q d) (p q) ( p r) a) p (q r) b) p p p q r q r p (q r) T T T T T T T F F F T F T F F T F F T T F T T T F F T F F T F F T F T F F F T F p p p p T F F F T T c) (p q) (p q) p q p q p q (p q) (p q) T T T T F T F F T T F T F T T F F F F F d) (p q) ( p r) p q r p q p r p r (p q) ( p r) T T T T F F T F T T F T F T T F T F T F F F T T T F F F F T T T F T T F T F F F F T F F T T T T F F T T T F F T F F F T T T T F Discrete Mathematics I Tutorial 1 4
5 7. Steve would like to determine the relative salaries of three coworkers using two facts. First, he knows that if Fred is not the highest paid of the three, then Janice is. Second, he knows that if Janice is not the lowest paid, then Maggie is paid the most. Is it possible to determine the relative salaries of Fred, Maggie, and Janice from what Steve knows? If so, who is paid the most and who the least? Explain your reasoning. p: if Fred is not the highest paid of the three, then Janice is q: if Janice is not the lowest paid, then Maggie is paid the most Fred Janice Maggie p q T F T T T F F T T F F T Fred > Maggie > Janice Discrete Mathematics I Tutorial 1 5
6 8. Five friends have access to a chat room. Is it possible to determine who is chatting if the following information is known? Either Kevin or Heather, or both, are chatting. Either Randy or Vijay, but not both, are chatting. If Abby is chatting, so is Randy. Vijay and Kevin are either both chatting or neither is. If Heather is chatting, then so are Abby and Kevin. Explain your reasoning. p: Either Kevin or Heather, or both, are chatting q: Either Randy or Vijay, but not both, are chatting r: If Abby is chatting, so is Randy s: Vijay and Kevin are either both chatting or neither is t: If Heather is chatting, then so are Abby and Kevin Heather Kevin Randy Vijay Abby p q r s t T T T F T T T T F T T F T F T T T T T F F T T F T T T T F T T T F T T T T F T T T F F T T T T F F F F T F T T T T F T T T T T F F T T T F F T F T F F T T T T F F T T F F T T T F T T T F T F T T T T F T F F T F T T T F F F T F T F T T T T T Chatting: Kevin and Vijay Not Chatting: Heather, Randy, Abby Discrete Mathematics I Tutorial 1 6
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3.4 THE CONDITIONAL & BICONDITIONAL Definition. Any statement that can be put in the form If p, then q, where p and q are basic statements, is called a conditional statement and is written symbolically
August 27, Review of Algebra & Logic. Charles Delman. The Language and Logic of Mathematics. The Real Number System. Relations and Functions
and of August 27, 2015 and of 1 and of 2 3 4 You Must Make al Connections and of Understanding higher mathematics requires making logical connections between ideas. Please take heed now! You cannot learn
Why Learning Logic? Logic. Propositional Logic. Compound Propositions
Logic Objectives Propositions and compound propositions Negation, conjunction, disjunction, and exclusive or Implication and biconditional Logic equivalence and satisfiability Application of propositional
Introduction Logic Inference. Discrete Mathematics Andrei Bulatov
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Introduction Propositional Logic. Discrete Mathematics Andrei Bulatov
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1 Propositional Logic
1 Propositional Logic Required reading: Foundations of Computation. Sections 1.1 and 1.2. 1. Introduction to Logic a. Logical consequences. If you know all humans are mortal, and you know that you are