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8/11/2019 1 Midterm EDISMAT (Logic and Propositional Calculus)
1/18
31
LOGIC ANDPROPOSITIONAL
CALCULUS
Adonis S. Santos, ECE
First Asia Institute of Technology and Humanities
8/11/2019 1 Midterm EDISMAT (Logic and Propositional Calculus)
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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
8/11/2019 1 Midterm EDISMAT (Logic and Propositional Calculus)
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LOGIC
Proof
PROPOSITIONAL CALCULUS
Logic using a formal system
Finite set of symbols
Grammar (syntax)
Set of rules
INTRODUCTION
8/11/2019 1 Midterm EDISMAT (Logic and Propositional Calculus)
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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
8/11/2019 1 Midterm EDISMAT (Logic and Propositional Calculus)
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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
8/11/2019 1 Midterm EDISMAT (Logic and Propositional Calculus)
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CONJUNCTION
p
q
BASIC LOGICAL OPERATIONS
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.
BASIC LOGICAL OPERATIONS
TRUE
FALSE
FALSE
FALSE
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DISJUNCTION
p
q
BASIC LOGICAL OPERATIONS
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.
BASIC LOGICAL OPERATIONS
TRUE
TRUE
TRUE
FALSE
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NEGATION
p
BASIC LOGICAL OPERATIONS
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
BASIC LOGICAL OPERATIONS
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
BASIC LOGICAL OPERATIONS
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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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