Discrete Mathematical Structures Course Details, Lecture notes of Mathematical Methods for Numerical Analysis and Optimization

Download Discrete Mathematical Structures Course Details and more Lecture notes Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity! CSL105: Discrete Mathematical Structures Ragesh Jaiswal, CSE, IIT Delhi Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Administrative Information Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Administrative Information Textbook: Discrete Mathematics and its Applications by Kenneth H. Rosen. Course webpage: http://www.cse.iitd.ac.in/~rjaiswal/2013/csl105. The site will contain course information, references, homework problems, tutorial problems, and announcements. Please check this page regularly. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Introduction Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Introduction What are Discrete Mathematical Structures? Discrete: Separate or distinct. Structures: Objects built from simpler objects as per some rules/patterns. Discrete Mathematics: Study of discrete mathematical objects and structures. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic: Propositional Logic Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic Why study logic in Computer Science? Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic Why study logic in Computer Science? Argue correctness of a computer program. Automatic verification. Check security of a cryptographic protocol. ... Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic Definition (Proposition) A proposition is a declarative statement (that is, a sentence that declares a fact) that is either true or false, but not both. Are these statements propositions? New Delhi is the capital of India. Yes. What time is it? No. Please read the first two sections of the book after this lecture. No. 2 + 2 = 5. Yes. x + 1 = 2. No. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic Definition (Proposition) A proposition is a declarative statement (that is, a sentence that declares a fact) that is either true or false, but not both. Propositional variable: Variables that represent propositions. Truth value: The truth value of a proposition is true (denoted by T) if it is a true proposition and false (denoted by F) if it is a false proposition. The area of logic that deals with propositions is called propositional logic or propositional calculus. Compound proposition: Proposition formed from existing proposition using logical operators. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic: logical operators Negation (¬): Let p be a proposition. The negation of p (denoted by ¬p), is the statement “it is not the case that p.” The proposition ¬p is read as “not p”. The truth value of the ¬p is the opposite of the truth value of p. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic: logical operators Negation (¬) Conjunction (∧): Let p and q be propositions. The conjunction of p and q (denoted by p ∧ q) is the proposition “p and q”. The conjunction p ∧ q is true when both p and q are true and is false otherwise. p q p ∧ q T T T T F F F T F F F F Table : Truth table for p ∧ q. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic: logical operators Negation (¬) Conjunction (∧) Disjunction (∨): Let p and q be propositions. The disjunction of p and q (denoted by p ∨ q) is the proposition “p or q”. The disjunction p ∨ q is false when both p and q are false and is true otherwise. p q p ∨ q T T T T F T F T T F F F Table : Truth table for p ∨ q. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic: logical operators Negation (¬) Conjunction (∧) Disjunction (∨). Exclusive or (⊕): Let p and q be propositions. The exclusive or of p and q (denoted by p⊕ q) is the proposition that is true when exactly one of p and q is true and is false otherwise. p q p⊕ q T T F T F T F T T F F F Table : Truth table for p ⊕ q. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic: logical operators Definition (Conditional statement) Let p and q be propositions. The conditional statement p → q is the proposition that is is false when p is true and q is false, and true otherwise. Definition (Converse) The converse of a proposition p → q is the proposition q → p. Definition (Contrapositive) The contrapositive of a proposition p → q is the proposition ¬q → ¬p. Definition (Inverse) The inverse of a proposition p → q is the proposition ¬p → ¬q. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic: logical operators Definition (Conditional statement) Let p and q be propositions. The conditional statement p → q is the proposition that is is false when p is true and q is false, and true otherwise. Definition (Converse) The converse of a proposition p → q is the proposition q → p. Definition (Contrapositive) The contrapositive of a proposition p → q is the proposition ¬q → ¬p. Definition (Inverse) The inverse of a proposition p → q is the proposition ¬p → ¬q. Show that the contrapositive of p → q always has the same truth value as p → q. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures Logic Propositional Logic: logical operators Definition (Conditional statement) Let p and q be propositions. The conditional statement p → q is the proposition that is is false when p is true and q is false, and true otherwise. Definition (Converse) The converse of a proposition p → q is the proposition q → p. Definition (Contrapositive) The contrapositive of a proposition p → q is the proposition ¬q → ¬p. Definition (Inverse) The inverse of a proposition p → q is the proposition ¬p → ¬q. Show that the contrapositive of p → q always has the same truth value as p → q. Show that, neither converse nor inverse of p → q has the same truth value as p → q for all truth values of p and q. Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures