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Prepared by

Dr. Nandhini S SAS

Logic is a study about reasoning

Validity of arguments Consistency among set of statements Matters of truth and falsehood

History of Logic

In the middle of the last century Boole laid the foundation for logic.

But real work of Mathematical logic was developed by Gottlob Frege, a German mathematician.

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Logic is the study of formal (i.e. symbolic) systems of reasoning.

Since the latter half of the twentieth century, logic has been used in computer science for various purposes ranging from program specification and verification to theorem-proving techniques.

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Statement or Proposition It is a declarative sentence which is either true or false but not both. We shall use the capital letters A,B,., P,Q(Except T and F) to represent statements. Truth value is the value returned by the statement (either TorF) Examples

Types of Statement Simple or atomic or primitive statement

Statements which do not contain any of the connectives are called as simple statements.

Compound or Molecular Statement Statements formed by combining two or more atomic statements with the connectives. Examples

Truth Table A table showing all possible truth values of a compound statement is called as a truth table.

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Negation - NOT Conjunction - AND Disjunction - OR Implication - IF.. THEN Bi-conditional IFF

Examples

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Wffs are constructed using the following rules: 1. True and False are wffs. 2. Each propositional constant (i.e. specific proposition), and

each propositional variable (i.e. a variable representing propositions) are wffs.

3. If A and B are wffs, then so are A, (A conjunction B), (A disjunction B), (A B). Tautology

a logical expression that is true for all variable assignments. Contradiction

a logical expression that is false for all variable assignments. Contingent

a logical expression that is neither a tautology nor a contradiction. Problems in class

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Let A & B be two statement formula. If the truth value of A is equal to the truth value of B for every one of the possible sets of truth values assigned, then A & B are said to be equivalent or logically equivalent denoted by A B.

Problems with truth table in the class

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Equivalence Name

pTp pFp Identity laws pTT pFF Domination laws ppp ppp Idempotent laws (p)p Double negation law pqqp pqqp Commutative laws

(pq)rp(qr) (pq)rp(qr) Associative laws

p(qr)(pq)(pr) p(qr)(pq)(pr) Distributive laws

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Equivalence Name

(pq)pq (pq)pq De Morgan's laws

p(pq)p p(pq)p Absorption laws

ppT ppF Negation laws

Note : T represents the true (T) for all possible assignments. F represents the false (F) for all possible assignments.

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Problems using results Practice Problems

Logical equivalences involving implication pqpq pqqp pqpq pq(pq) (pq)pq (pq)(pr)p(qr) (pq)(pr)p(qr) (pr)(qr)(pq)r (pr)(qr)(pq)r Logical equivalences involving biconditionals pq(pq)(qp) pqpq pq(pq)(pq) (pq)pq

Normal forms are alternate procedures to prove a statement formula a tautology or contradiction.

Disjunctive normal forms A formula which is equivalent to a given formula and which consists of a sum of elementary products. Conjunctive normal forms

A formula which is equivalent to a given formula and which consists of a product of elementary sums.

Problems discussed in the class

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Principal Disjunctive normal forms (PDNF) A formula which is equivalent to a given formula and which consists of a disjunction of minterms only is called as PDNF. Also called as sum-of-products canonical form. Principal Conjunctive normal forms

A formula which is equivalent to a given formula and which consists of a conjunction of maxterms only is called as PCNF. Also called as product-of sums canonical form.

Problems discussed in the class

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