The goal of this course is to introduce students to ideas and techniques from discrete mathematics that are widely used in computer science. A relation between two sets a and b is a sub set of axb and is denoted by r. What are the applications of relation in discrete mathematics. A set is a collection of objects, called elements of the set. The section contains multiple choice questions and answers on.
Presents propositional logic, combinatorics, methods of proof, mathematical systems, algebra of sets, matrix algebra, relations and functions, recursion and generating functions, applications to computer science, and graph theory. We have provided multiple complete discrete mathematical structures notes pdf for any university student of bca, mca, b. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. Cs103 handout 16 summer 2019 july 10, 2019 guide to proofs on. Ix x i prove that the identity function is a bijection. Cse 1400 applied discrete mathematics relations and functions. Inverse functions i every bijection from set a to set b also has aninverse function i the inverse of bijection f, written f 1, is the function that assigns to b 2 b a unique element a 2 a such that fa b i observe. Thus symmetry and transitivity together imply reflexivity. Includes elementary logic and set theory, equivalence relations, functions, counting arguments, asymptotic complexity, inductively defined sets, recursion, graphs and trees, boolean algebra and combinatorial circuits, finite state automata, and diagonalization and countability. Covers mathematical structures that naturally arise in computer science. So far, weve seen simple structures like sets and natural numbers, but there are many other important structures out. The set theory was founded by georg cantor in late nineteenth century. In these discrete mathematics handwritten notes pdf, we will study the fundamental concepts of sets, relations, and functions, mathematical logic, group theory, counting theory, probability, mathematical induction, and recurrence relations, graph theory, trees and boolean algebra.
Solve problems relating to sets, functions and relations. Examples of discrete structures built with the help of sets. R is transitive x r y and y r z implies x r z, for all x,y,z. Preface this book is designed for a one semester course in discrete mathematics for sophomore or junior level students. Discrete mathematics virginia commonwealth university.
Mathematical structures just as there are common data structures in programming, there are common mathematical structures in discrete math. Cartesian product csci 1900 discrete structures cartesian. Sets, functions, relations northwestern university. Algorithms and growth of functions pdf, docx lecture 9. Discrete structures lecture notes stanford university. Sets, relations and functions quiz attention jee main standard aspirants. A binary relation from a to b is a subset of a cartesian product a x b. Functions assume a and b are nonempty sets a function f from a to b is an assignment of exactly one element of b to each element of a fa b if f assigns b to a f. Cartesian product csci 1900 discrete structures cartesian product. A complements of a set a is the set of elementsthingsobjects which are not in set a. The domain x is the set of elements that appear on the lefthand.
It covers sets, logic, proving techniques, combinatorics, functions, relations, graph theory and algebraic structures. These are the lecture notes of the course discrete structures 2it50. A function defined on sets a,b a b assigns to each. These basic concepts of sets, logic functions and graph theory are applied to boolean algebra and logic networks while the advanced concepts of functions and algebraic structures are applied to finite state machines and coding. Many different systems of axioms have been proposed. Problem set two introduced firstorder logic and gave you some practice writing more intricate proofs than before.
Discrete mathematics forms the mathematical foundation of computer and information science. Many important discrete structures are built using sets, which are collections of objects. Just exam provide a platform to all students who want to make practice for various subject online. I this is why bijections are also calledinvertible functions instructor. Sets and notation, induction, proof techniques, divisibility, prime numbers, modular arithmetic, relations and functions, mathematical logic, counting, binomial coefficients, the inclusionexclusion principle, the pigeonhole principle, asymptotic notation, graphs, trees, planar graphs. Basic concepts of set theory, functions and relations. Students are assumed to have taken comp 1805 discrete structures i, which covers mathematical reasoning, basic proof techniques, sets, functions, relations, basic graph theory, asymptotic notation, and countability. The emphasis is on the applications of discrete structures in computer science rather than the mathematical theory itself.
X the inverse relation r of a relation r is defined as. It contains well written, well thought and well explained computer science and programming articles, quizzes and practicecompetitive programmingcompany interview questions. The empty relation between sets x and y, or on e, is the empty set the full relation between sets x and y is the set x. Theres something like 7 or 8 other types of relations. A oneterm introduction to discrete structures with applications to computing problems. Discrete mathematics for computer science department of. Discrete mathematics mcq multiple choice questions. Basic concepts of set theory, functions and relations 1. Basic building block for types of objects in discrete mathematics. Discrete mathematics is the branch of mathematics dealing with objects that can consider only distinct, separated values. The emphasis is on the applications of discrete structures in computer science rather than the.
A course in discrete structures cornell university. Relation example a is the set of all students and b is the set of all courses a relation r may be defined as the course is required paul giblock r csci 2710 danny camper r csci 2710 csci 1900 discrete structures relations page 9 relations across same set relations may be from one set to the same set. R tle a x b means r is a set of ordered pairs of the form a,b where a a and b b. This video contains the description about unit2 notes of discrete mathematics. The union of a and b, denoted by a b, is the set that contains those elements that are either in. Discrete mathematics questions and answers sanfoundry. Hauskrecht basic discrete structures discrete math study of the discrete structures used to represent discrete objects many discrete structures are built using sets sets collection of objects examples of discrete structures built with the help of sets.
Relations page 5 of 35 35 more generally, for any sets a and b, we can form their cartesian product in accordance with the following definition. Discrete mathematics mcq multiple choice questions with introduction, sets theory, types of sets, set operations, algebra of sets, multisets, induction, relations, functions and algorithms etc. Discrete mathematics tutorial provides basic and advanced concepts of discrete mathematics. Cantors diagonal argument to show the reals are uncountable. Edaa40 discrete structures in computer science seminar 2 a. A set a is a subset of a set b iff every element of a is also an element of b. Now, about the applications of set relations in speci. Relation example a is the set of all students and b is the set of all courses a relation r may be defined as the course is required paul giblock r csci 2710 danny camper r csci 2710 csci 1900 discrete structures relations page 9 relations across same set relations may be from one set to the same set, i. This chapter will be devoted to understanding set theory, relations, functions. Discrete mathematics functions 1646 bijection example i theidentity function i on a set a is the function that assigns every element of a to itself, i. Subsets a set a is a subset of a set b iff every element of a is also an element of b. The sets could be discrete or continuous which is concerned with the way sets are arranged, counted or combined. Topics include sets, relations, functions, counting, graphs, trees, recursion, propositional and predicate logic, proof techniques, boolean algebra. Cse 1400 applied discrete mathematics relations department of.
Discrete structures lecture notes by vladlen koltun. Discrete structures provide a tool box in computer science. Then identify whether each is a function, onto function, onetoone function, bijection. Relations page 4 of 35 35 before continuing, we note that the following notation is also common in the literature. R tle a x b means r is a set of ordered pairs of the form a,b. Number theory a the math behind the rsa crypto system. This course introduces the applications of discrete mathematics in the field of computer science. Learners will become familiar with a broad range of mathematical objects like sets, functions, relations, graphs, that are omnipresent in computer science.
This course will roughly cover the following topics and speci c applications in computer science. Sets, relations, and functions introduction sets relations functions summary problems find, read. Now that youve hit problem set three, youll be com. Tech branch to enhance more knowledge about the subject and to score. Discrete mathematics representing relations geeksforgeeks. A successful discrete mathematics course should carefully blend and balance all.
Question bank sets, relations and functions just exam. The concept of a set appears in all mathematical structures. Partee 1979, fundamentals of mathematics for linguistics. The zoom meeting id for our online lectures is 931 0753 5633 you can join using this link we will be using discord to run office hours, and it will also serve as our peer support and qa forum. Sets are one of the basic building blocks for the types of objects considered in discrete mathematics important for counting programming languages have set operations set theory is an important branch of mathematics many different systems of axioms have been used to develop set theory. Two elements a and b listed in a specific order pair, denoted by a,b two order pairs a,b and c,d are said to be equal iff ac and bd relation in a set. Discrete structures in computer science seminar 2 a recap. Introduction sets relations functions summary problems. Issues about data structures used to represent sets and the computational cost of set operations.
During a 12week term with three hours of classes per week, i cover most. Lecture notes for college discrete mathematics download book. Guide to proofs on discrete structures in problem set one, you got practice with the art of proofwriting in general as applied to numbers, sets, puzzles, etc. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department.
Sets, functions, sequences, sums and matrices much of discrete mathematics is devoted to the study of discrete structures, used to represent discrete objects. These basic concepts of sets, logic functions and graph theory are applied to boolean algebra and logic networks while the advanced concepts of functions and algebraic structures are applied to finite state machines and coding theory. This definition involves a generalized set abstract. Sets, relations, functions introduction basic objects in.
We study topics in such areas as sets, logic, proof techniques, induction procedures, relations, functions, graphs, trees, combinatorics, and recursive procedures. Discrete structure cs302 b tech rgpv aicte flexible. Reflexive, symmetric, transitive, and antisymmetric relations. These are the mathematical equivalent of data structures, the way. Direct and inverse image of a set under a relation. Our discrete mathematics structure tutorial is designed for beginners and professionals both. Edaa40 discrete structures in computer science seminar 2 a recap.
Number theory athe math behind the rsa crypto system. The cardinality of a finite set is the number of elementsthingsobjects in that set. Sets and set operations cs 441 discrete mathematics for cs m. Given a set a and an equivalence relation on a, for any we define the equivalence class of a as given a set a and an equivalence relation on a, the quotient set. Introduction sets are one of the basic building blocks for the types of objects considered in discrete mathematics important for counting programming languages have set operations set theory is an important branch of mathematics many different systems of axioms have been used to develop set theory here we are not concerned with a formal set of axioms for. A set can be represented by listing its elements between braces. Sets, relations and functions, sequences, sums, cardinality of sets richard mayr. Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart. This book is designed for a one semester course in discrete mathematics.
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