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Discrete Mathematics by Himanshu Gulati

Himanshu Gulati
Himanshu Gulati

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Note for Discrete Mathematics - DMS by HIMANSHU GULATI


by Himanshu Gulati

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UNIT-I: Propositional Logic 1. Introduction to Logic: Logic: logic comprises a (formal) language for making statements about objects and reasoning about properties of these objects. Statements in a logical language are constructed according to a predefined set of formation rules (depending on the language) called syntax rules. Logic languages are used instead of natural languages as natural languages are very vast so cannot be formally described. Also, natural languages are ambiguous, context sensitive and verbose. A logical system , or a “logic” for short, typically consists of three things (but may consist of only the first two, or the first and third) 1. A syntax , or set of rules specifying what expressions are part of the language of the system, and how they may be combined to form more complex expressions/statements (often called “formulæ)”. 2. A semantics , or set of rules governing the meanings or possible meanings of expressions, and how the meaning, interpretation, evaluation and truth value of complex expressions depend on the meaning or interpretation of the parts. 3. A deductive system , or set of rules governing what makes for an acceptable or endorsed pattern of reasoning within in the system 1.1. Propositional Logic: • • • • • • • • Propositional logic is the system of logic with the simplest semantics. Many of the concepts and techniques used for studying propositional logic generalize to first -order logic. In propositional logic, there are atomic assertions (or atoms, or propositional letters) and compound assertions built up from the atoms and the logical connectives, and, or, not, implication and equivalence. The atomic facts are interpreted as being either true or false. In propositional logic, once the atoms in a proposition have received an interpretation, the truth value of the proposition can be computed. Technically, this is a consequence of the fact that the set of propositions is a freely generated inductive closure. Certain propositions are true for all possible interpretations. They are called tautologies. Intuitively speaking, a tautology is a universal truth. Hence, tautologies play an important role. A proposition is a statement that can be either true or false o “The sky is blue” o “I is a English major” o “x == y” Not propositions: o “Are you Bob?” o “x = 7” 1.1.1. Syntax and Semantic of Propositional Logic Here we will give a purely syntactic definition of propositional logic. • Statements of this language are propositional formulas. HIMANSHU GULATI- Foundation of Computer Science