**Mathematical Foundations of Computer** **Science**

## OBJECTIVES:

• To introduce the scholars to the topics and techniques of separate ways and

combinatorial reasoning.

• To introduce a large style of applications. The recursive approach to the answer of

problems is prime in separate arithmetic, and this approach reinforces the shut

ties between this discipline and also the space of engineering.

### UNIT -I:

Mathematical Logic: Propositional Calculus: Statements and Notations, Connectives, Well

Formed Formulas, Truth Tables, Tautologies, Equivalence of Formulas, Duality Law,

Tautological Implications, traditional Forms, Theory of logical thinking for Statement Calculus,

Consistency of Premises, Indirect methodology of Proof. Predicate Calculus:Predicative Logic,

Statement Functions, Variables and Quantifiers, Free and certain Variables, logical thinking Theory for

Predicate Calculus.

### UNIT -II:

Set Theory: Introduction, Operations on Binary Sets, Principle of Inclusion and Exclusion,

Relations: Properties of Binary Relations, Relation Matrix and alphabetic character, Operations on Relations,

Partition and Covering, transitive Closure, Equivalence, Compatibility and Partial Ordering

Relations, Hasse Diagrams, Functions: Bijective Functions, Composition of Functions, Inverse

Functions, Permutation Functions, algorithmic Functions, Lattice and its Properties.

### UNIT- III:

Algebraic Structures and variety Theory: pure mathematics Structures:Algebraic Systems,

Examples, General Properties, Semi teams and Monoids, similarity of Semi teams and

Monoids, Group, Subgroup, Abelian group, similarity, isomorphy, Number

Theory:Properties of Integers, Division Theorem, the best common measure, Euclidean

Algorithm, Least integer, Testing for Prime Numbers, the elemental Theorem of

Arithmetic, standard Arithmetic (Fermat’s Theorem and Euler’s Theorem)

### UNIT -IV:

Combinatorics: Basic of numeration, Permutations, Permutations with Repetitions, Circular

Permutations, Restricted Permutations, mixtures, Restricted mixtures, Generating

Functions of Permutations and mixtures, Binomial and Multinomial Coefficients, Binomial

and Multinomial Theorems, The Principles of Inclusion–Exclusion, Pigeonhole Principle and its

Application.

II Year – I Semester

L T P C

4 0 0 3

### UNIT -V:

Recurrence Relations: Generating Functions, operate of Sequences, Partial Fractions,

Calculating constant of Generating Functions, return Relations, Formulation as

Recurrence Relations, determination return Relations by Substitution and Generating Functions,

Method of Characteristic Roots, determination nonuniform return Relations

### UNIT -VI:

Graph Theory: Basic ideas of Graphs, Sub graphs, Matrix illustration of Graphs:

Adjacency Matrices, Incidence Matrices, similarity Graphs, methods and Circuits, Eulerian and

Hamiltonian Graphs, Multigraphs, tabular Graphs, Euler’s Formula, Graph Colouring and

Covering, Chromatic variety, Spanning Trees, Algorithms for Spanning Trees (Problems solely

and Theorems while not Proofs).

### OUTCOMES:

• Student are going to be ready to demonstrate skills in determination mathematical issues

• Student are going to be ready to comprehend mathematical principles and logic

• Student are going to be ready to demonstrate data of mathematical modeling and

proficiency in victimization mathematical computer code

• Student are going to be ready to manipulate and analyze information numerically and/or diagrammatically victimization

appropriate computer code

• Student are going to be ready to communicate effectively mathematical ideas/results verbally or in

writing

### TEXT BOOKS:

1.Discrete Mathematical Structures with Applications to engineering, J. P. Tremblay

and P. Manohar, Tata McGraw Hill.

- components of separate Mathematics-A laptop orientating Approach, C. L. Liu and D. P.

Mohapatra, 3rdEdition, Tata McGraw Hill. - separate arithmetic and its Applications with Combinatorics and Graph Theory, K. H.

Rosen, seventh Edition, Tata McGraw Hill.

REFERENCE BOOKS: - separate arithmetic for laptop Scientists and Mathematicians, J. L. Mott, A. Kandel,

T.P. Baker, second Edition, novice Hall of India. - separate Mathematical Structures, BernandKolman, Robert C. Busby, Sharon dealer

Ross, PHI. - separate arithmetic, S. K. Chakraborthy and B.K. Sarkar, Oxford, 2011.