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

Lecture № 2 Fundamentals of Logic

Outline  Logical Form and Logical Equivalence  Logical Equivalence; Tautologies and Contradictions;  Summary of Logical Equivalences  Conditional Statements  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;

Playing with implication 10/23/2015

Playing with implication 10/23/2015

Truth Table of equivalence 10/23/2015

Logically Equivalent Statements 10/23/2015

Tautologies 10/23/2015

Contradictions 10/23/2015

Logic Equivalences 10/23/2015 or

Why this is important? 10/23/2015

Examples 10/23/2015

Examples 10/23/2015

Equivalence and Tautology 10/23/2015

Example 10/23/2015

Examples 10/23/2015

Truth Tables 10/23/2015 pq pqpqqpqppqpq pp qq pqpq  (p  q)  p  q FFFFTTTTTT FTTTTTFTFF TFTTFFTFFF TTTTTFFTFF

Double Negation 10/23/2015

DeMorgan’s Laws 10/23/2015 pq pqpqpqpq pp qq  p  q  (p  q)  (p  q)  p  q FFFFTTTTTT FTTFTFTFTF TFTFFTTFTF TTTTFFFFFF

Example  Write in C++ the condition for floating point variable size, in which the message “Present” will be displayed for the following code fragment if ( size > 25 || size == 19 ) cout<<”Future”; else if ( size 2) cout<<"Past"; else cout <<"Present"; 10/23/2015

“Algebraic” Laws of Logic 10/23/2015

“Logic” Laws of Logic 10/23/2015

Number of Rows in Truth Table 10/23/2015

Expressing Connectives 10/23/2015

Example 10/23/2015

Example 10/23/2015

Example: Decreasing number of comparisons 10/23/2015

Example: Decreasing number of comparisons 10/23/2015

Example 10/23/2015

Example 10/23/2015

Logic Inference

First Law of Substitution 10/23/2015

Second Law of Substitution 10/23/2015

Logic inference 10/23/2015

Inference and Tautology 10/23/2015

General Definition of Inference 10/23/2015

Examples  Tautology  X   X  Logical Contradiction  X   X  Negation of Tautology  Valid – if all values are true – the logic value is true  X  Y  X  Y  Logic Equivalence 10/23/2015 and

Inference Rules  “Modus ponens"  "Modus tollens” 10/23/2015

Modus Ponens 10/23/2015

Rule of Tollens 10/23/2015

Example 10/23/2015

MORE READING: CHAPTER 2 SUSANNA S. EPP, DISCRETE MATHEMATICS WITH APPLICATIONS Questions? 10/23/2015