1 midterm edismat (logic and propositional calculus)

8/11/2019 1 Midterm EDISMAT (Logic and Propositional Calculus)

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31

LOGIC ANDPROPOSITIONAL

CALCULUS

Adonis S. Santos, ECE

First Asia Institute of Technology and Humanities


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LOGIC AND PROPOSITIONAL CALCULUS

Introduction

Proposition and Compound Statements

Basic Logical Operations

Propositions and Truth Tables

Tautologies and Contradictions

Logic Equivalence

Algebra of Propositions

Conditional and Biconditional Statements

Arguments

Propositional Functions, Quantifiers

Negation of Quantified Statements

OUTLINE


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LOGIC

Proof

PROPOSITIONAL CALCULUS

Logic using a formal system

Finite set of symbols

Grammar (syntax)

Set of rules

INTRODUCTION


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PROPOSITION Statement)

Declarative statement which is TRUE or F

but not both

(a) Ice floats in water. (b) China is in

(c) 2 + 2 = 4 (d) 2 + 2 = 5

(e) Where are you going? (f) Do your ho

Only the first four are propositions

(a) and (c) are TRUE, but (b) and (d) are FALSE

PROPOSITIONS AND COMPOUND STATE


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COMPOUND STATEMENTS Composite)

Composed of subpropositions with variou

connectives

(g) Roses are red and violets are blue(h)John is smart or he studies every nigh

Primitive cannot be broken down into s

propositions

PROPOSITIONS AND COMPOUND STATE

CONNECTIVES


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CONJUNCTION

Compound proposition combined with and

DISJUNCTION

Compound proposition combined with or

NEGATION

Putting not in a proposition

Truth value depends only on the truth values

propositions

BASIC LOGICAL OPERATIONS


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CONJUNCTION

p

q


p and q

If p and q are true,

then p

q is true;

otherwise p q is false.


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CONJUNCTION: EXAMPLE

Ice floats in water and 2 + 2 = 4.

Ice floats in water and 2 + 2 = 5.

China is in Europe and 2 + 2 = 4.

China is in Europe and 2 + 2 = 5.


TRUE

FALSE

FALSE

FALSE


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DISJUNCTION

p

q


p or q

If p and q are false,

then p

q is false;

otherwise p q is true.


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DISJUNCTION: EXAMPLE

Ice floats in water or 2 + 2 = 4.

Ice floats in water or 2 + 2 = 5.

China is in Europe or 2 + 2 = 4.

China is in Europe or 2 + 2 = 5.


TRUE

TRUE

TRUE

FALSE


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NEGATION

p


not p

If p is true, then p is false;and if p is false, then p is true.


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NEGATION: EXAMPLE

It is false that ice floats in water.

Ice does not float in water.

It is false that 2 + 2 = 5.

2 + 2 5


FALSE

FALSE

TRUE

TRUE


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LOGICAL NOTATION

AND

p q q pq p

OR

p + q p q

NOT

p' p ~p p



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Proposition are constructed with logical varithat take on the value of TRUE or FALSE with

connectives

(p q)

PROPOSITIONS AND TRUTH TAB


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TAUTOLOGIES

Propositions whose truth values in the tru

are all TRUE

CONTRADICTIONS

Propositions whose truth values in the truare all FALSE

TAUTOLOGIES ANG CONTRADICTI


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TAUTOLOGIES CONTRADICTIO

One is the negation of the other

TAUTOLOGIES ANG CONTRADICTI


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Two propositions arelogically equivalent

if tthe same truth table.

LOGICAL EQUIVALENCE

(p q) p q


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LECTURE LOGIC AND PROPOSITIONAL CALCULUS

Introduction

Proposition and Compound Statements

Basic Logical Operations

Propositions and Truth Tables

Tautologies and Contradictions Logic Equivalence

Algebra of Propositions

Conditional and Biconditional Statements

Arguments

Propositional Functions, Quantifiers

Negation of Quantified Statements

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1 midterm edismat (logic and propositional calculus)

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