Logic

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Propositional Logic

​ TABLE OF CONTENTS

​1) Propositions - Slide 3

​2) Basic logical operations and conditional & biconditional statements - Slide 5

​(a) Conjunction - Slide 6

​(b) Disjunction - Slide 8

​(c) Negation - Slide 12

​(d) Conditional Statement - Slide 14

​(e) Biconditional Statement - Slide

​(f) Converse, Inverse, Contrapositive - Slide 17

​3) Truth Tables - Slide 20

​4) Logically Equivalent

​5) Laws of the Algebra of Propositions

​1. Propositions

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A proposition (a statement or verbal assertion) is a declarative sentence (a sentence that declares a fact) that is either True (T/1) or False (F/0), but not both. Prepositions will normally be denoted by p, q, r, ...

The truth or falsity of a proposition is called its truth value , i.e T or F / 1 or 0.

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​Sentences that are Propositions

​(a) Paris is in France (b) 1 + 1 = 2 ​(c) Mary has a pen (d) 3 + 5 = 10

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​Sentences that are not propositions

​(a) Where are u going? (b) Do your homework. (c) Close the door.

​Many propositions are composite, that is, composed of sub propositions and various logical connectives, such as "and" , "or" and "negation." Such composite propositions are called compound propositions.

​The fundamental property of a compound proposition is that its truth value is completely determined by the truth values of its sub proposition together with the way in which they are connected to form the compound proposition.

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​Sentences that are compound propositions

​(a) Roses are red and violets are blue.

​(b) Hilary is beautiful or she is poor.

​**(c) Soup or salad is served with your meal.

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2. BASIC LOGICAL OPERATIONS AND CONDITIONAL OR BICONDITIONAL STATEMENTS

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Example :

​For the statement below, write the converse, the inverse and the contrapositive statements.

​"If it is raining, then the streets are wet."

​Let p : It is raining, then not p : It is not raining

​Let q : The streets are wet, then not q : The streets are not wet.

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​Converse : If q, then p - If the streets are wet, then it is raining.

​Inverse : If not p, then not q - If it is not raining, then the streets are not wet.

​Contrapositive : If not q, then not p - If the streets are not wet, then it is not raining.

3) Truth Tables

​Example of a Tautology

​Example of a Contradiction

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​Example of a Contingency

​Logically Equivalent

​5) Laws of the Algebra of Propositions

​continued

​continued

​Examples

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​3.

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​***5. (Absorption Law)

​***6. (Absorption Law)