COS 150 Discrete Structures Assoc. Prof. Svetla Boytcheva Fall semester 2014

Lecture № 1 Fundamentals of Logic

Code of Ethics  All course materials are adapted version of the textbook: Susanna S. Epp, Discrete Mathematics with Applications, Fourth Edition, Cengage Learning.  Some images – McGrawHill  For materials from other sources, please see the copyright reverence below each slide. 9/17/2015

Use of Logic  Propositional Logic  First Order Logic (Quantifiers)/ Predicate Logic  Boolean Algebra 9/17/2015

Outline  Logical Form and Logical Equivalence  Statements; Compound Statements; Truth Values; Evaluating the Truth of More General  Compound Statements; Logical Equivalence; Tautologies and Contradictions;  Summary of Logical Equivalences  Conditional Statements  Logical Equivalences Involving →; Representation of If-Then As Or ;  The Negation of a Conditional Statement;  The Contrapositive of a Conditional Statement;  The Converse and Inverse of a Conditional Statement; Only If and the Biconditional;  Necessary and Sufficient Conditions;  Valid and Invalid Arguments  Modus Ponens and Modus Tollens;  Additional Valid Argument Forms: Rules of Inference;

Use of Logic  In mathematics  Give a precise meaning of statements  Distinguish between valid and invalid arguments  Provide use of “correct” reasoning  Natural language can be very ambiguous  He ate the cookies on the couch  This is a good soup  You could do with a new automobile. How about a test drive? 9/17/2015

Use of Logic  Natural language can be very ambiguous  This is a good soup  You could do with a new automobile. How about a test drive?  I shot an elephant in my pajamas. 9/17/2015

Use of Logic  In computing  Design new data/knowledge from existing fact  Design of computer circuits  Construction of computer programs  Verification of correctness of programs and circuit design  Specification 9/17/2015

Statements (propositions)  Propositional logic deals with statements and their truth value  Truth values are TRUE (T or 1) and FALSE (F or 0) 9/17/2015

Example Statements  1+1= 2 (statement, T)  The moon is made of cheese (statement, F)  Go home! (no statement, imperative)  What a beautiful garden (no statement, exclamation)  Alice said: “What a beautiful garden ” (statement, depends on Alice)  Y+1=2 (no statement, uncertain) 9/17/2015

Logic connectives 9/17/2015

Logic connectives 9/17/2015

Compound Statements 9/17/2015

Order of Operations  9/17/2015

Example  9/17/2015

Translating from English to Symbols: But and Neither-Nor  “Jim is tall but he is not heavy.”  Shakespeare: “Neither a borrower nor a lender be” 9/17/2015

And, Or, and Inequalities 9/17/2015

Truth Tables 9/17/2015

Truth Table of negation  Unary connective p: “Today is Wednesday”  p: “Today is not Wednesday” 9/17/2015

 Binary connective  Example p: “Today is Wednesday” q: “It is raining” p  q: “Today is Wednesday and it is raining” Truth Table of conjunction 9/17/2015

Truth Table of disjunction 9/17/2015  Binary connective  Example p: “Today is Friday” q: “Today is Saturday” p  q: “Today is Friday or Saturday”

 Binary connective  Example p  q “You can follow the rules or be disqualified” Truth Table of exclusive or 9/17/2015

Truth Table of implication  Binary connective  Example p -> q: “If black is white, then we live in Antarctica” 9/17/2015

Implication as a promise 9/17/2015

MORE READING: CHAPTER 2 SUSANNA S. EPP, DISCRETE MATHEMATICS WITH APPLICATIONS Questions? 9/17/2015