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COS 150 Discrete Structures Assoc. Prof. Svetla Boytcheva Fall semester 2014
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Lecture № 2 Fundamentals of Logic
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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;
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Playing with implication 10/23/2015
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Playing with implication 10/23/2015
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Truth Table of equivalence 10/23/2015
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Logically Equivalent Statements 10/23/2015
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Tautologies 10/23/2015
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Contradictions 10/23/2015
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Logic Equivalences 10/23/2015 or
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Why this is important? 10/23/2015
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Examples 10/23/2015
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Examples 10/23/2015
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Equivalence and Tautology 10/23/2015
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Example 10/23/2015
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Examples 10/23/2015
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Truth Tables 10/23/2015 pq pqpqqpqppqpq pp qq pqpq (p q) p q FFFFTTTTTT FTTTTTFTFF TFTTFFTFFF TTTTTFFTFF
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Double Negation 10/23/2015
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DeMorgan’s Laws 10/23/2015 pq pqpqpqpq pp qq p q (p q) (p q) p q FFFFTTTTTT FTTFTFTFTF TFTFFTTFTF TTTTFFFFFF
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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
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“Algebraic” Laws of Logic 10/23/2015
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“Logic” Laws of Logic 10/23/2015
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Number of Rows in Truth Table 10/23/2015
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Expressing Connectives 10/23/2015
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Example 10/23/2015
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Example 10/23/2015
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Example: Decreasing number of comparisons 10/23/2015
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Example: Decreasing number of comparisons 10/23/2015
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Example 10/23/2015
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Example 10/23/2015
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Logic Inference
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First Law of Substitution 10/23/2015
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Second Law of Substitution 10/23/2015
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Logic inference 10/23/2015
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Inference and Tautology 10/23/2015
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General Definition of Inference 10/23/2015
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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
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Inference Rules “Modus ponens" "Modus tollens” 10/23/2015
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Modus Ponens 10/23/2015
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Rule of Tollens 10/23/2015
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Example 10/23/2015
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MORE READING: CHAPTER 2 SUSANNA S. EPP, DISCRETE MATHEMATICS WITH APPLICATIONS Questions? 10/23/2015
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