# Discrete Mathematical Structure

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# Code(Credit) : CUTM1522(3-1-2)

## Course Objectives

• To understand mathematical reasoning in order to read, comprehend and construct mathematical arguments as well as to solve problems, occurred in the development of programming languages.
• To work with different types of Sets, Lattices and Boolean Algebra.

## Learning Outcomes

Upon successful completion of this course, students will be able to :

• Evaluate elementary mathematical arguments and identify fallacious reasoning.
• Apply the logical structure of proofs and work symbolically with connectives and quantifiers to produce logically valid, correct and clear arguments.
• Reformulate statements from common language to formal logic. Apply truth tables and the rules of propositional and predicate calculus.

## Course Syllabus

Module-I    ( T-5 hrs-P-4 hrs-P-4 hrs )

Introduction to Logic, Logical connectives, Converse, Inverse, Contrapositive, Proportional Equivalences, Propositional Calculus, Translating English sentences to Logical expressions.

Practice-1  : Finding the truth values of compound propositions using Python

Practice-2  :  Writing  conditional statements in Python

Project-1  :  Study on equivalent statements in Logic

Project-2  :  Study on Propositional Calculus

Module-II   ( T-5 hrs-P-0 hrs-P-4 hrs )

Predicates, Quantifiers, Bound and free variables, Rules of inference, Derivation of Valid conclusion, Nested Quantifiers.

Project-3  :  Study on Universal Quantifiers and Existential Quantifiers

Project-4  :  Study on Valid arguments in Logic

Module-III    ( T-4 hrs-P-0 hrs-P-4 hrs )

Recurrence Relation and it's solution by using Generating function, Pigeon-Hole Principle.

Project-5  :  Study on different types of Recurrence relations

Project-6  :  Study on Pigeon-Hole Principle

Module-IV    ( T-4 hrs-P-6 hrs-P-2 hrs )

The basics of counting, Generalized Permutations, Combinations and their Applications.

Practice-3  :  Finding Factorial of a number using Python

Practice-4  :  Finding out the value of P(n,r) using Python

Practice-5  :  Finding out the value of C(n,r) using Python

Project-7  :  Study on permutations and combinations

Module-V    ( T-4 hrs-P-2 hrs-P-2 hrs )

Relations and their operations and properties, Equivalence relation, Representation of a relation, Closure of a relation.

Practice-6  :  Finding Composition of Relations using Python

Project-8  :  Study on different types of Relations

Module-VI   ( T-3 hrs-P-0 hrs-P-4 hrs )

Partial order relation, Partially ordered set and totally ordered set, Hasse diagram, Maximal and minimal elements, Greatest and least elements, Supremum and infimum of a Poset.

Project-9  :  Study on partially ordered sets and Totally ordered sets

Project-10  :  Study on Hasse Diagram

Module-VII    ( T-5 hrs-P-0 hrs-P-4 hrs )

Lattice and it's algebraic structure, Properties of a Lattice, De-Morgan's Law, Bounded and Distributive Lattices, Boolean Algebra, Boolean Function.

Project-11  :  Study on Lattices and Ordered Sets

Project-12  :  Study on Boolean Algebra

Text Book:

1. Kenneth H. Rosen, Discrete Mathematics and Applications, Tata Mc-Graw Hill Publications,

Chapters: 1(1.1 To 1.5), 5(5.1, 5.2, 5.5), 6(6.1, 6.2, 6.4), 7, 10(10.1, 10.2).

Reference Book:

1. Discrete Mathematics with Graph Theory by Edgar G. Goodaire and Michael M. Parmenter

## Session 4

Proportional Equivalence, Definitions of Tautology Contradiction and Contingency

https://slideplayer.com/slide/8280241/

## Session 6 & 7

Practice-1

Finding the truth values of compound propositions using Python

## Session 8 & 9

Practice-2

Writing  conditional statements in Python

## Session 10 & 11

Project-1

Study on equivalent statements in Logic

## Session 12 & 13

Project-2

Study on Propositional Calculus

## Session 14

Predicates and Quantifiers definitions Universal and Existential Quantifiers

https://slideplayer.com/slide/760301/

## Session 19 & 20

Project-3

Study on Universal Quantifiers and Existential Quantifiers

## Session 21 & 22

Project-4

Study on Valid arguments in Logic

## Session 27 & 28

Project-5

Study on different types of Recurrence relations

## Session 29 & 30

Project-6

Study on Pigeon-Hole Principle

## Session 35 & 36

Practice-3

Finding Factorial of a number using Python

## Session 37 & 38

Practice-4

Finding out the value of P(n,r) using Python

## Session 39 & 40

Practice-5

Finding out the value of C(n,r) using Python

## Session 41 & 42

Project-7

Study on permutations and combinations

## Session 47 & 48

Practice-6

Finding Composition of Relations using Python

## Session 49 & 50

Project-8

Study on different types of Relations

## Session 53

Maximal and minimal element, Supremum and infimum of a Poset

## Session 54 & 55

Project-9

Study on partially ordered sets and Totally ordered sets

## Session 56 & 57

Project-10

Study on Hasse Diagram

## Session 63 & 64

Project-11

Study on Lattices and Ordered Sets

## Session 65 & 66

Project-12

Study on Boolean Algebra