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

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CSL105: Discrete Mathematical Structures

Ragesh Jaiswal, CSE, IIT Delhi

Administrative Information

Administrative Information

Grading Scheme (^1) Quizzes: 4 quizzes, 2 points each. (^2) Homework: 6 homework, 5 points each. (^3) Minor 1 and 2: 15 points each. (^4) Major: 30 points. (^5) Attendance: 2 points. Policy on late submission of homework: Homework should be submitted in the beginning of the lecture on the deadline. You will lose 25% of the points per day for late submissions. Policy on cheating: Anyone found using unfair means in the course will receive an F grade. You must write homework solutions on your own.

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.

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.

Introduction

Why study Discrete Mathematics? Information processing and computation may be interpreted as manipulation of discrete structures. Enable you to think logically and argue about correctness of computer programs and analyze them. What you should expect to learn from this course: Rigorous thinking! Mathematical foundations of Computer Science.

Logic: Propositional Logic

Logic

Propositional Logic

Why study logic in Computer Science?

Logic

Propositional Logic

Why study logic in Computer Science? Argue correctness of a computer program. Automatic verification. Check security of a cryptographic protocol. ... Propositional logic: Basic form of 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.

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. What time is it? Please read the first two sections of the book after this lecture. 2 + 2 = 5. x + 1 = 2.

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.

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.

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. Examples: p: Tigers have been seen in this area. ¬p: It is not the case that a tiger has been seen in this area.

p ¬p T F F T Table : Truth table for ¬p.

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.