csl105: discrete mathematical structures
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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
Instructor
Ragesh JaiswalOffice: 515, Bharti BuildingEmail: rjaiswal@cse.iitd.ac.in
Teaching Assistants
Suyash Roongta (email: cs5090531@cse.iitd.ernet.in)Utkarsh Ohm (email: cs5090256@cse.iitd.ac.in)Davis Issac (email: csz138106@cse.iitd.ac.in)Ankit Anand (email: csz138105@cse.iitd.ac.in)
Please send email to set up a meeting with me or the TAs.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
Administrative Information
Grading Scheme1 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 lectureon 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 anF grade.You must write homework solutions on your own.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
Administrative Information
Textbook: Discrete Mathematics and its Applications byKenneth H. Rosen.
Course webpage:http://www.cse.iitd.ac.in/~rjaiswal/2013/csl105.
The site will contain course information, references, homeworkproblems, tutorial problems, and announcements. Please checkthis 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 somerules/patterns.
Discrete Mathematics: Study of discrete mathematical objectsand structures.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
Introduction
Why study Discrete Mathematics?
Information processing and computation may be interpreted asmanipulation of discrete structures.Enable you to think logically and argue about correctness ofcomputer programs and analyze them.
What you should expect to learn from this course:
Rigorous thinking!Mathematical foundations of Computer Science.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
Introduction
Topics:
Logic: propositional logic, predicate logic, proofs.mathematical induction etc.Fundamental Structures: sets functions, relations, recursivefunctions etc.Counting: Pigeonhole principle, permutation and combination,recurrence relations, generating functions, inclusion-exclusionetc.Graphs: representing graphs, connectivity, shortest paths etc.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
Logic: Propositional Logic
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
LogicPropositional Logic
Why study logic in Computer Science?
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
LogicPropositional 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
LogicPropositional 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 thatdeclares a fact) that is either true or false, but not both.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
LogicPropositional Logic
Definition (Proposition)
A proposition is a declarative statement (that is, a sentence thatdeclares 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.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
LogicPropositional Logic
Definition (Proposition)
A proposition is a declarative statement (that is, a sentence thatdeclares 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
LogicPropositional Logic
Definition (Proposition)
A proposition is a declarative statement (that is, a sentence thatdeclares 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 (denotedby T) if it is a true proposition and false (denoted by F) if itis a false proposition.
The area of logic that deals with propositions is calledpropositional logic or propositional calculus.
Compound proposition: Proposition formed from existingproposition using logical operators.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
LogicPropositional 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
LogicPropositional 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: A Tiger has been seen in this area.¬p: ?
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
LogicPropositional 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 ¬pT F
F T
Table : Truth table for ¬p.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
LogicPropositional Logic: logical operators
Negation (¬)
Conjunction (∧): Let p and q be propositions. Theconjunction of p and q (denoted by p ∧ q) is the proposition“p and q”. The conjunction p ∧ q is true when both p and qare true and is false otherwise.
p q p ∧ qT 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
LogicPropositional Logic: logical operators
Negation (¬)
Conjunction (∧)
Disjunction (∨): Let p and q be propositions. The disjunctionof 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 andis true otherwise.
p q p ∨ qT 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
LogicPropositional Logic: logical operators
Negation (¬)
Conjunction (∧)
Disjunction (∨).
Exclusive or (⊕): Let p and q be propositions. The exclusiveor of p and q (denoted by p⊕ q) is the proposition that is truewhen exactly one of p and q is true and is false otherwise.
p q p⊕ qT 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
LogicPropositional Logic: logical operators
Negation (¬)Conjunction (∧)Disjunction (∨).Exclusive or (⊕)Conditional statement (→): Let p and q be propositions. Theconditional statement p → q is the proposition “if p, then q.” Theconditional statement p → q is false when p is true and q is false,and true otherwise. In the conditional statement p → q, p is calledthe hypothesis (or antecedent or premise) and q is called theconclusion (or consequence).
p q p→ qT T T
T F F
F T T
F F T
Table : Truth table for p → q.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
LogicPropositional Logic: logical operators
Definition (Conditional statement)
Let p and q be propositions. The conditional statement p → q is theproposition that is is false when p is true and q is false, and trueotherwise.
q is true on the condition that p is true.This is also called an implication.There are various ways to express p → q:
“p is sufficient for q” or “a sufficient condition for q is p”“q if p”“q when p”“q is necessary for p” or “a necessary condition for p is q”“q unless ¬p”“p implies q”“p only if q”“q whenever p”“q follows from p”
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
LogicPropositional Logic: logical operators
Definition (Conditional statement)
Let p and q be propositions. The conditional statement p → q is theproposition that is is false when p is true and q is false, and trueotherwise.
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
LogicPropositional Logic: logical operators
Definition (Conditional statement)
Let p and q be propositions. The conditional statement p → q is theproposition that is is false when p is true and q is false, and trueotherwise.
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 truthvalue as p → q.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
LogicPropositional Logic: logical operators
Definition (Conditional statement)
Let p and q be propositions. The conditional statement p → q is theproposition that is is false when p is true and q is false, and trueotherwise.
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 truthvalue as p → q.Show that, neither converse nor inverse of p → q has the sametruth value as p → q for all truth values of p and q.
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
End
Ragesh Jaiswal, CSE, IIT Delhi CSL105: Discrete Mathematical Structures
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