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Chapter 1 SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAs

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### CHAPTER OUTLINE: PART III

1.3.1 INTRODUCTION

1.3.2 PROPOSITION

1.3.3 COMPOUND STATEMENTS

1.3.4 LOGICAL CONNECTIVES

1.3.5 CONDITIONAL STATEMENT

1.3.6 PROPOSITIONAL EQUIVALENCES

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

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.

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.

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

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

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

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

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)

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 ( )

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 ( )

Converse, Contrapositive

CONVERSE : the implication of is called converse of

CONTRAPOSITIVE : the contrapositive of is the implication

Example: refer textbook

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.

1.3.6 PROPOSITIONAL EQUIVALENCES

Example:

1.3.6 PROPOSITIONAL EQUIVALENCES

Contingency

- A proposition that is neither a tautology nor a contradiction

Example: refer text book

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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