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Chapter 1 SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAs

Chapter 1 SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAs. BY: MISS FARAH ADIBAH ADNAN IMK. 1.3 ELEMENTARY LOGIC. 1.3.1 INTRODUCTION. 1.3.2 PROPOSITION. CHAPTER OUTLINE: PART III. 1.3.3 COMPOUND STATEMENTS. 1.3.4 LOGICAL CONNECTIVES. 1.3.5 CONDITIONAL STATEMENT.

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Chapter 1 SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAs

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  1. Chapter 1SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAs BY: MISS FARAH ADIBAH ADNAN IMK

  2. 1.3 ELEMENTARY LOGIC 1.3.1 INTRODUCTION 1.3.2 PROPOSITION CHAPTER OUTLINE: PART III 1.3.3 COMPOUND STATEMENTS 1.3.4 LOGICAL CONNECTIVES 1.3.5 CONDITIONAL STATEMENT 1.3.6 PROPOSITIONAL EQUIVALENCES

  3. 1.3 ELEMENTARY LOGIC 1.3.1 INTRODUCTION • Logic – used to distinguish between valid and invalid mathematical arguments. • Application in computer science – design computer circuits, construction of computer program, verification of the correctness of programs. • Basic building blocks - Prepositions

  4. 1.3.2 PROPOSITION • Proposition – is a declarative sentence either true or false, but not both. • Eg: • Washington, D.C., is the capital of the United States of America. • 1 + 1 = 2 • What time is it? • Read this carefully. • x + 1 = 2 • Letters are used to denote prepositions – p, q, r, s.

  5. 1.3.3 COMPOUND STATEMENTS • Many mathematical statements are constructed by combining one or more propositions. Eg: John is smart or he studies every night. • Fundamental property of a compound proposition: The truth value is determined by the truth value of its subpropositions, together with the way they are connected to form compound proposition.

  6. 1.3.4 LOGICAL CONNECTIVES 1) Not (negation) : ~ / Let p be a proposition. The negation of p is denoted by , and read as “not p”. -Eg: Find the negation of the preposition “Today is Friday”. The Truth Table for the Negation of a Preposition

  7. 1.3.4 LOGICAL CONNECTIVES 2) And (conjunction) : Let p and q be prepositions. The preposition of “p and q” - denoted , is TRUE when BOTH p and q are true and otherwise is FALSE. The Truth Table for the Conjunction of Two Prepositions

  8. 1.3.4 LOGICAL CONNECTIVES 3) Or (disjunction) : Let p and q be prepositions. The preposition of “p or q” - denoted , is FALSE when BOTH p and q are FALSE and TRUE otherwise. The Truth Table for the Disjunction of Two Prepositions

  9. EXAMPLE 1.1 Consider the following statements, and determine whether it is true or false. • Ice floats in water and 2 + 2 = 4 • China is in Europe and 2 + 2 = 4 • 5 – 3 = 1 or 2 x 2 = 4

  10. EXAMPLE 1.2 Let p and q be the following propositions: p = It is below freezing q = It is snowing Translate the following into logical notation, using p and q and logical connectives. • It is below freezing and snowing • It is below freezing but not snowing • It is not below freezing and it is not snowing • It is either snowing or below freezing (or both)

  11. 1.3.5 CONDITIONAL STATEMENTS 1) Conditional Statement/ Implication Let p and q be a preposition. The implication is the preposition that is FALSE when p is true, q is false. Otherwise is TRUE. p = hypothesis/antecedent/premise q = conclusion/consequence Express: “ if p, then q”, “q when p”, “p implies q” The Truth Table for the Implication ( )

  12. 1.3.5 CONDITIONAL STATEMENTS 2) Equivalence/ Biconditional Let p and q be a preposition. The biconditional is the preposition that is TRUE when p and q have the same truth values, and FALSE otherwise. Express: “ p if and only if q” The Truth Table for the Biconditional ( )

  13. Converse, Contrapositive CONVERSE : the implication of is called converse of CONTRAPOSITIVE : the contrapositive of is the implication Example: refer textbook

  14. 1.3.6 PROPOSITIONAL EQUIVALENCES Tautology • A compound proposition that is always TRUE, no matter what the truth values of the propositions that occur in it. • Contains only “T” in the last column of their truth table. Contradiction • A compound proposition that is always FALSE. • Contains only “F” in the last column of their truth table.

  15. 1.3.6 PROPOSITIONAL EQUIVALENCES Example:

  16. 1.3.6 PROPOSITIONAL EQUIVALENCES Contingency • A proposition that is neither a tautology nor a contradiction Example: refer text book

  17. 1.3.6 PROPOSITIONAL EQUIVALENCES Logically Equivalent Two propositions p and q are said to be logically equivalent, or simply equivalent or equal, denoted by if they have identical truth tables. Example: Find the truth tables of

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