# Discrete Mathematics By Swapan Kumar Chakraborty -Discrete Mathematics By S K Sarkar

**Discrete Mathematics By Swapan Kumar Chakraborty**: Discrete Mathematics Is Designed To Serve As A Textbook For Undergraduate Engineering Students Of Computer Science And Postgraduate Students Of Computer Applications. The Book Would Also Prove Useful To Post Graduate Students Of Mathematics. The Book Seeks To Provide A Thorough Understanding Of The Subject And Present Its Practical Applications To Computer Science.

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Discrete Mathematics (Oxford Higher Education Book by S. Chakraborty and B.K. Sarkar.

Book Published in 2011.

Book explains the all Concepts of Mathematical Foundation for Computer Science.

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Beginning with an overview of basic concepts like Sets, Relation and Functions, and Matrices, the book delves into core concepts of discrete mathematics like

*Combinatorics,*

*Logic and Truth Tables,*

*Groups,*

*Order Relation and Lattices,*

*Boolean Algebra,*

*Trees and Graphs.*

Special emphasis is also laid on certain advanced topics like Complexity and Formal Language and Automata.

**Discrete Mathematics Problems And Solutions**

Algorithms and programmers have been used wherever required to illustrate the applications.

Written in a simple, student-friendly style, the book provides numerous solved examples and chapter end exercises to help students apply the mathematical tools to computer-related concepts.

**Discrete Mathematics Book By Swapan Kumar Chakraborty:**

Discrete Mathematics is a comprehensive and lucid book for undergraduate and postgraduate students of Computer Science Engineering. The book comprises of chapters on the language of mathematics, techniques, algorithms and graphs, and algebraic methods. In addition, the book contains 1000 individual exercises for thorough revision and final practice. This book is essential for computer science engineers preparing for GATE.

**About Oxford University Press:**

Oxford University Press is a renowned publishing house that develops and publishes high quality textbooks, scholarly works, and academic books for school courses, bilingual dictionaries and also digital materials for both learning and teaching. It is a division of the University of Oxford. The first book was locally published in the year 1912. Some of the books published under their banner are India’s Ancient Past, Atkins’ Physical Chemistry, Companion to Politics in India, Sociology: Themes and Perspectives and Common Mistakes at IELTS Advanced …and How to Avoid Them.

**About Author:**

S.K. Chakraborty is currently Associate Professor at the Department of Applied Mathematics, BIT, Mesra, Ranchi. An alumnus of IIT Kharagpur, he holds a Ph D in applied mathematics from the Indian School of Mines, Dhanbad and has about 20 years of academic experience. He has guided many research scholars and published several research papers in various journals of national and international repute. His research interest includes seismology, rock mechanics, earthquake engineering, and heat transfer. He is also a member of the Indian Society for Theoretical and Applied Mechanics, and the Indian National Science Congress Association.

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N B.K. Sarkar is currently Assistant Professor at the Department of Information Technology, BIT, Mesra, Ranchi. An M Phil in computer science from Annamalai University, he did his MCA from BESU (Howrah) and MSc (Mathematics) from IIT Kharagpur. He has more than 10 years of teaching experience, and is a life member of the Indian Society for Technical Education. His research areas include data mining, parallel computing, evolutionary computing.

**Swapan kumar sarkar discrete mathematics s. Chand 4th ed. 2006:**

Elements of Discrete Mathematics is a comprehensive book undergraduate students of Computer Science Engineering. The book comprises chapters on sets and permutations, permutations, combinations and discrete probability, relations and functions, graphs and planar graphs, modeling computation, analysis of algorithms and recurrence relations and recursive algorithms. In addition, the book consists of several solved examples, practice problems, illustrations and algorithms to understand the concepts better. This book is essential for candidates preparing for various competitive examinations like GATE and IES.

**Discrete Mathematics Lecture Notes ** A set can be written explicitly by listing its elements using set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set.

Some Example of Sets

- A set of all positive integers
- A set of all the planets in the solar system
- A set of all the states in India
- A set of all the lowercase letters of the alphabet

Representation of a Set

Sets can be represented in two ways −

- Roster or Tabular Form
- Set Builder Notation

Roster or Tabular Form

The set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas.

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**Example 1** − Set of vowels in English alphabet, A={a,e,i,o,u}A={a,e,i,o,u}

**Example 2** − Set of odd numbers less than 10, B={1,3,5,7,9}B={1,3,5,7,9}

Set Builder Notation

**Example 2**− The set {1,3,5,7,9}{1,3,5,7,9} is written as −

B={x:1≤x<10 and (x%2)≠0}B={x:1≤x<10 and (x%2)≠0}

If an element x is a member of any set S, it is denoted by x∈Sx∈S and if an element y is not a member of set S, it is denoted by y∉Sy∉S.

**Example** − If S={1,1.2,1.7,2},1∈SS={1,1.2,1.7,2},1∈S but 1.5∉S1.5∉S

Some Important Sets

**N** − the set of all natural numbers = {1,2,3,4,…..}{1,2,3,4,…..}

**Z** − the set of all integers = {…..,−3,−2,−1,0,1,2,3,…..}{…..,−3,−2,−1,0,1,2,3,…..}

**Z ^{+}** − the set of all positive integers

**Q** − the set of all rational numbers

**R** − the set of all real numbers

**W** − the set of all whole numbers

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Cardinality of a Set

Cardinality of a set S, denoted by |S||S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞∞.

**Example** − |{1,4,3,5}|=4,|{1,2,3,4,5,…}|=∞|{1,4,3,5}|=4,|{1,2,3,4,5,…}|=∞

If there are two sets X and Y,

- |X|=|Y||X|=|Y| denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ from X to Y.
- |X|≤|Y||X|≤|Y| denotes that set X’s cardinality is less than or equal to set Y’s cardinality. It occurs when number of elements in X is less than or equal to that of Y. Here, there exists an injective function ‘f’ from X to Y.
- |X|<|Y||X|<|Y| denotes that set X’s cardinality is less than set Y’s cardinality. It occurs when number of elements in X is less than that of Y. Here, the function ‘f’ from X to Y is injective function but not bijective.
- If |X|≤|Y|If |X|≤|Y| and |X|≤|Y||X|≤|Y| then |X|=|Y||X|=|Y|. The sets X and Y are commonly referred as equivalent sets.

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Types of Sets

Sets can be classified into many types.

A set X is a subset of set Y (Written as X⊆YX⊆Y) if every element of X is an element of set Y.

**Example 1** − Let, X={1,2,3,4,5,6}X={1,2,3,4,5,6} and Y={1,2}Y={1,2}.

. A Set X is a proper subset of set Y (Written as X⊂YX⊂Y) if every element of X is an element of set Y and |X|<|Y||X|<|Y|.

**Example** − Let, X={1,2,3,4,5,6}X={1,2,3,4,5,6} and Y={1,2}Y={1,2}. Here set Y⊂XY⊂Xsince all elements in YY are contained in XX too and XX has at least one element is more than set YY.

Universal Set

It

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In this case, set of all mammals is a subset of UU, set of all fishes is a subset of UU, and set of all insects is a subset of UU, and so on.

Empty Set or Null Set

An empty set contains no elements. It is denoted by ∅∅. As the number of elements in an empty set is finite, empty set is a finite set. The cardinality of empty set or null set is zero.

**Example** − S={x|x∈NS={x|x∈N and 7<x<8}=∅7<x<8}=∅

Singleton Set or Unit Set

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Singleton set or unit set contains only one element. A singleton set is denoted by {s}{s}.

**Example** − S={x|x∈N, 7<x<9}S={x|x∈N, 7<x<9} = {8}{8}

Equal Set

If two sets contain the same elements they are said to be equal.

**Example** − If A={1,2,6}A={1,2,6} and B={6,1,2}B={6,1,2}, they are equal as every element of set A is an element of set B and every element of set B is an element of set A.

Equivalent Set

If the cardinalities of two sets are same, they are called equivalent sets.

**Example** − If A={1,2,6}A={1,2,6} and B={16,17,22}B={16,17,22}, they are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A|=|B|=3|A|=|B|=3

Overlapping Set

Two sets that have at least one common element are called overlapping sets.

In case of overlapping sets −

- n(A∪B)=n(A)+n(B)−n(A∩B)n(A∪B)=n(A)+n(B)−n(A∩B)
- n(A∪B)=n(A−B)+n(B−A)+n(A∩B)n(A∪B)=n(A−B)+n(B−A)+n(A∩B)
- n(A)=n(A−B)+n(A∩B)n(A)=n(A−B)+n(A∩B)
- n(B)=n(B−A)+n(A∩B)n(B)=n(B−A)+n(A∩B)

**Example** − Let, A={1,2,6}A={1,2,6} and B={6,12,42}B={6,12,42}. There is a common element ‘6’, hence these sets are overlapping sets.

Disjoint Set