BY: MISS FARAH ADIBAH ADNANBY: MISS FARAH ADIBAH ADNAN

IMKIMK

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

1.3 ELEMENTARY LOGIC1.3 ELEMENTARY LOGIC

1.3.1 INTRODUCTION1.3.1 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

1.3.2 PROPOSITION1.3.2 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 = 23) What time is it?4) Read this carefully.5) x + 1 = 2 Letters are used to denote prepositions – p,

q, r, s.

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 STATEMENTS1.3.3 COMPOUND STATEMENTS

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 CONNECTIVES1.3.4 LOGICAL CONNECTIVES

p

p

T F

F T

p

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 CONNECTIVES1.3.4 LOGICAL CONNECTIVES

p q

p q

T T T

T F F

F T F

F F F

p q

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 CONNECTIVES1.3.4 LOGICAL CONNECTIVES

p q

T T T

T F T

F T T

F F F

p q

p q

EXAMPLE 1.1EXAMPLE 1.1

Consider the following statements, and determine whether it is true or false.1)Ice floats in water and 2 + 2 = 42)China is in Europe and 2 + 2 = 43)5 – 3 = 1 or 2 x 2 = 4

EXAMPLE 1.2EXAMPLE 1.2

Let p and q be the following propositions:p = It is below freezingq = 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)

1) Conditional Statement/ ImplicationLet 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 STATEMENTS1.3.5 CONDITIONAL STATEMENTS

p q

T T T

T F F

F T T

F F T

p q

p q

p q

2) Equivalence/ BiconditionalLet 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 STATEMENTS1.3.5 CONDITIONAL STATEMENTS

p q

T T T

T F F

F T F

F F T

p q

p q

p q

CONVERSE : the implication of is called converse of

CONTRAPOSITIVE : the contrapositive of is the implication

Example: refer textbook

CONVERSE, CONTRAPOSITIVECONVERSE, CONTRAPOSITIVE

p qq p

p q q p

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 EQUIVALENCES1.3.6 PROPOSITIONAL EQUIVALENCES

Example:

1.3.6 PROPOSITIONAL EQUIVALENCES1.3.6 PROPOSITIONAL EQUIVALENCES

T F T F

F T T F

pp p p p p

Contingency A proposition that is neither a tautology nor a

contradiction

Example: refer text book

1.3.6 PROPOSITIONAL EQUIVALENCES1.3.6 PROPOSITIONAL EQUIVALENCES

Logically EquivalentTwo 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 EQUIVALENCES1.3.6 PROPOSITIONAL EQUIVALENCES

p q

( ) and pp q q

p q p^q -(p^q) p q -p -q -p v -q

T T T F T T F F F

T F F T T F F T T

F T F T F T T F T

F F F T F F T T T