Discrete Mathematics

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College Book: SPECTRUM(4TH - Sem)

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The book introduction and Syllabus:      Download

Module 1: 

Section 1 (Set Theory)                                           Download

Section 2 (Functions)                                  Download

Section 3 (Functions)                                   Download

Section 4 (Countable And Uncountable Sets)             Download

Section 5(Principles of  mathematical induction)         Download

Module 2:

1 Permutation And Combinations                   Download

2 Pigeonhole Principle                            Download

3 The Inclusion-Exclusion Principle           Download

Module 3:

1 Logic and propositional Calculus                       Download

Module 4:

1 Algebraic Structures And Morphism                Download

2 Algebraic Structure With Two Binary Operations, Rings, integral Domain And Fields                                               Download

3 Boolean Algebra                                                  Download

    

Module 5:

1 Graph Theory                       Download          

2 Trees                                      Download

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ASSIGNMENT 1:                                    Download

ASSIGNMENT 1 Solution:                     Download       

Assignment No. 1

 

Subject and Subject code: Discrete Mathematics (BTCS 401-18)                     Semester: 4th

 

The date on which the assignment was given: 24 /02/2021              Date of submission of assignment:02/03/2021 Total Marks:10

Course Outcomes

CO1	To be able to express logical sentence in terms of predicates, quantifiers, and logical connectives
CO2	To derive the solution for a given problem using deductive logic and prove the solution based on logical inference
CO3	For a given a mathematical problem, classify its algebraic structure
CO4	To evaluate Boolean functions and simplify expressions using the properties of Boolean algebra
CO5	To develop the given problem as graph networks and solve it with techniques of graph theory

 

Sr. No.	Assignment related to COs	Marks(10)	Relevance to CO No.
Q1.	(a)  State and prove De Morgan’s Law. (b)  Prove that if P, Q, R, S are arbitrary sets, then (P ∩ Q) × (R ∩ S) = (P ´ R) ∩ (Q ´ S) (c)  Let A, B, C are any sets, such that A È B = A ÈC and A Ç B = A ÇC then show that B = C	1+1+1	CO1
Q2.	How many integers between 1 and 300 (inclusive) are: (i)  Divisible by at least one of 3, 5, 7? (ii)  Divisible by 3 and 5, not by 7? (iii)  Divisible by 5 but neither by 3 nor by 7?	1+1+1	CO3
Q3.	(a) Define Equivalence Relation with example. (b)(i) Prove that R is symmetric iff R-1 =R. (ii) If R is an equivalence relation on a set A, then so R-1. (c) Let R be a relation defined on the set of real nos. a is related to b if a ≤ b where a, b are real numbers. Then R is a Partial order relation.	1+1+1+1	CO1

 

Assignment 1 Solution:                                   Download

SYLLABUS (Detailed Contents):

Module 1: 

Sets, Relation, and Function: Operations and Laws of Sets, Cartesian Products, Binary

Relation, Partial Ordering Relation, Equivalence Relation, Image of a Set, Sum, and Product of

Functions, Bijective functions, Inverse and Composite Function, Size of a Set, Finite and

infinite Sets, Countable and uncountable sets, Cantor's diagonal argument, and The Power Set

theorem, Schroeder-Bernstein theorem.

Principles of Mathematical Induction: The Well-Ordering Principle, Recursive definition,

The Division algorithm: Prime Numbers, The Greatest Common Divisor: Euclidean Algorithm,

The Fundamental Theorem of Arithmetic. 

Module 2: 

Basic counting techniques-inclusion and exclusion, pigeon-hole principle, permutation, and combination.

Module 3: 

Propositional Logic: Syntax, Semantics, Validity and Satisfiability, Basic Connectives and

Truth Tables, Logical Equivalence: The Laws of Logic, Logical Implication, Rules of

Inference, The use of Quantifiers. Proof Techniques: Some Terminology, Proof Methods

and Strategies, Forward Proof, Proof by Contradiction, Proof by Contraposition, Proof of

Necessity and Sufficiency. 

Module 4: 

Graphs and Trees:  Graphs and their properties, Degree, Connectivity, Path, Cycle, Sub

Graph, Isomorphism, Eulerian and Hamiltonian Walks, Graph Colouring, Colouring maps and

Planar Graphs, Colouring Vertices, Colouring Edges, List Colouring, Perfect Graph, definition

properties and Example, rooted trees, trees and sorting, weighted trees and prefix codes, Bi-

connected component and Articulation Points, Shortest distances.