Math.3336: Discrete Mathematics. Applications of Propositional Logic

Transcription

1 Math.3336: Discrete Mathematics Applications of Propositional Logic Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston blerina Fall 2018 Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 1/16

2 Assignments to work on Homework #1 due Wednesday, 8/29, 11:59pm No credit unless turned in by 11:59pm on due date Late submissions not allowed, but lowest homework score dropped when calculating grades Homework will be submitted online in your CASA accounts. You can find the instructions on how to upload your homework in our class webpage. Read Sections 1.4, 1.5, 1.6 for next week. Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 2/16

3 Overview to Part I - The Propositional Logic Propositional Logic Summary The Language of Propositions Connectives Truth Values Truth Tables Applications Translating English Sentences System Specifications Logic Puzzles/Circuits Logical Equivalences Important Equivalences Showing Equivalence Satisfiability Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 3/16

4 Section 1.1 Review of the Basic Concepts A proposition is a statement that is either true or false. Truth value of a proposition identifies whether a proposition is true (written T) or false (written F) More complex propositions can be formed by using logical connectives (also called boolean connectives) The five basic logical connectives: 1 : negation (read not ) 2 : conjunction (read and ), 3 : disjunction (read or ), 4 : implication (read implies ) 5 : biconditional (read if and only if ) Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 4/16

5 Section 1.1 Review of the Basic Concepts(continued) Propositions formed using these logical connectives are called compound propositions; otherwise atomic propositions A propositional formula is either an atomic or compound proposition. Truth table for propositional formula F shows truth value of F for every possible value of its constituent atomic propositions. Two propositions are equivalent if they always have the same truth value. A truth table can be used to determine the equivalence of the given propositions. Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 5/16

6 Summary Conditional is of the form p q Converse: q p Inverse: p q Contrapositive: q p Conditional and contrapositive have same truth value Inverse and converse always have same truth value Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 6/16

7 Operator Precedence Given a formula p q r, do we parse this as (p q) r or p (q r)? Without settling on a convention, formulas without explicit paranthesization are ambiguous. To avoid ambiguity, we will specify precedence for logical connectives. Operator precedence is a convention that tells us how to parse formulas if they are not explicitly paranthesized. Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 7/16

8 Operator Precedence, cont. Negation ( ) has higher precedence than all other connectives. Question: Does p q mean (i) (p q) or (ii) ( p) q? Conjunction ( ) has next highest precedence. Question: Does p q q mean (i) (p q) r or (ii) p (q r)? Disjunction ( ) has third highest precedence. Next highest is precedence is, and lowest precedence is Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 8/16

9 Operator Precedence Example Which is the correct interpretation of the formula p q r q r (A) ((p (q r)) q) ( r) (B) ((p q) r) q) ( r) (C) (p (q r)) (q ( r)) (D) (p ((q r) q)) ( r) Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 9/16

10 Section Applications of Propositional Logic Translate English sentences into a propositional formula. System Specifications Logic Puzzles/Circuits Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 10/16

11 Different Ways of Expressing p q if p, then q if p, q p implies q q unless p q if p p only if q q when(ever) p q follows from p q is necessary for p p is sufficient for q a necessary condition for p is q a sufficient condition for q is p Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 11/16

12 Converting English into Logic Steps to convert an English sentence to a propositional formula: Identify atomic propositions and represent them using propositional variables. Determine appropriate logical connectives. Let p = I major in CS and q = I will find a good job. How do we translate following English sentences into logical formulas? If I major in CS, then I will find a good job : p q I will not find a good job unless I major in CS : p q It is sufficient for me to major in CS to find a good job : p q It is necessary for me to major in CS to find a good job : p q Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 12/16

13 More English - Logic Conversions Let p = I major in CS, q = I will find a good job, r = I can program. How do we translate following English sentences into logical formulas? I will not find a good job unless I major in CS or I can program : ( p r) q I will not find a good job unless I major in CS and I can program : ( p r) q A prerequisite for finding a good job is that I can program : r q If I major in CS, then I will be able to program and I can find a good job : p (r q) Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 13/16

14 System specifications System and Software engineers take requirements in English and express them in a precise specification language based on logic. Example: Express in propositional logic: The automated reply cannot be sent when the file system is full. Solution: One possible solution: Let p denote The automated reply can be sent and q denote The file system is full. Then we get q p. Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 14/16

15 Logic Puzzles An island has two kinds of inhabitants, knights, who always tell the truth, and knaves, who always lie. You go to the island and meet A and B. A says: B is a knight. B says: We are of opposite types. What are types of A and B? Solution: Let p and q be the statements that A is a knight and B is a knight, respectively. So, then p represents the proposition that A is a knave and q that B is a knave. If A is a knight, then p is true. Since knights tell the truth, q must also be true. Then (p q) ( p q) would have to be true, but it is not. So, A is not a knight and therefore p must be true. If A is a knave, then B must not be a knight since knaves always lie. Thus both p and q hold since both are knaves. Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 15/16

16 Logic Puzzles (continued) We will give the solution by using a truth table: Let p and q be the statements that A is a knight and B is a knight, respectively. Then: A says: B is a knight i.e. q B says: We are of opposite types i.e. (p q) ( p q) What are types of A and B? p q A: q B:(p q) ( p q) Result T T T F Not possible T F F T Not possible F T T T Not possible F F F F Answer Thus, both A and B must be knaves. Instructor: Dr. Blerina Xhabli, University of Houston Math.3336: Discrete Mathematics Applications of Propositional Logic 16/16