1. BY: MISS FARAH ADIBAH ADNAN IMK

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

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: 1) Washington, D.C., is the capital of the United States of America. 2) 1 + 1 = 2 3) What time is it? 4) Read this carefully. 5) x + 1 = 2 Letters are used to denote prepositions – p, q, r, s.

5. 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. 1.3.3 COMPOUND STATEMENTS

6. 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 1.3.4 LOGICAL CONNECTIVES p TF FT

7. 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 1.3.4 LOGICAL CONNECTIVES pq TTT TFF FTF FFF

8. 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 1.3.4 LOGICAL CONNECTIVES pq TTT TFT FTT FFF

9. EXAMPLE 1.1 Consider the following statements, and determine whether it is true or false. 1) Ice floats in water and 2 + 2 = 4 2) China is in Europe and 2 + 2 = 4 3) 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. (a) It is below freezing and snowing (b) It is below freezing but not snowing (c) It is not below freezing and it is not snowing (d) It is either snowing or below freezing (or both)

11. 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 ( ) 1.3.5 CONDITIONAL STATEMENTS pq TTT TFF FTT FFT

12. 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 ( ) 1.3.5 CONDITIONAL STATEMENTS pq TTT TFF FTF FFT

13. CONVERSE : the implication of is called converse of CONTRAPOSITIVE : the contrapositive of is the implication Example: refer textbook C ONVERSE, C ONTRAPOSITIVE

14. 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. 1.3.6 PROPOSITIONAL EQUIVALENCES

15. Example: 1.3.6 PROPOSITIONAL EQUIVALENCES TFTF FTTF

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

17. 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 1.3.6 PROPOSITIONAL EQUIVALENCES pqp^q-(p^q)pq-p-q-p v -q TTTFTTFFF TFFTTFFTT FTFTFTTFT FFFTFFTTT