Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
UNIT IS: Multiple Choice Questions
Lectures in Discrete Mathematics, Course 1, Bender/Williamson
Review Questions
Multiple Choice Questions for Review
In each case there is one correct answer (given at the end of the problem set). Try
to work the problem
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
[5] CL5: The last line of Example 2 should capitalize North and South.
[5] CL5: In the last sentence of Example 3, word should be name.
[27] CL27: The running head should be justied right, not centered.
*[27] CL27: Exercise 36: Should say An example
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Basic Concepts in Graph Theory
UNIT GT: Multiple Choice Questions
Lectures in Discrete Mathematics, Course 2, Bender/Williamson
Multiple Choice Questions for Review
Some of the following questions assume that you have done the exercises.
1. Indicate which
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
A Short Course in Discrete Mathematics
errata as of 11/8/2006
[page numbers] in Dover edition.
More important errors are marked with an asterisk.
[8] BF8: In Exercise 1.2, statements should be statement.
[15] BF15: In the middle of the paragraph that st
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
partial order
Partially Ordered Set (Poset)
A relation R on a set A is called
a partial ordering or partial
order if it is reflexive, antisymmetric, and transitive. Set
A together with a partial
ordering R is called a partially
ordered set, and is denoted
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Rules of inferences open
propositions with quantifiers
x : A(x) C(x)
x : C(x) M(x)
x : A(x)M(x)
Rules of inferences open
propositions with quantifiers
x : A(x) C(x)
x : C(x) M(x)
x : A(x)M(x)
Rules of inferences open
propositions with quantifiers
x :
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, Pilani
Pilani Campus
INSTRUCTION DIVISION
FIRST SEMESTER 201415
COURSE HANDOUT(PARTII)
Ausgust 1, 2014
In addition to PartI (General Handout for all courses appended to the Timetable) this portion gives furthe
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Representing Relations
special way to represent relations:
With a zeroone matrix.
R is a relation from A = cfw_u1, u2, , um
to C=cfw_q1, q2, , qn
Represented by the matrix MR
=[mij]mn
mij = 1 if (ui, qj) R
mij = 0 if (ui, qj) R
Example
A = cfw_1, 1,
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Logic
Logic = the study of correct
reasoning
Sentences
Sentences are usually classified as
declarative, exclamatory, interrogative
or imperative. We confine our attention
to those declarative sentences to
which we assign one and only one of
its truth val
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Lattices and boolean
algebra
Properties of lattices
Before proving a number of properties of
lattices we recall the meaning of x y
and xy
(i) xx y & yx y i.e x y is an upper
bound of x &y.
(ii) If xz & yz then x yz
i.e x y is least upper bound of x &
y x
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Fall 2014
BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI
INSTRUCTION DIVISION
FIRST SEMESTER 201314
Proposed COURSE HANDOUT (PARTII)
In addition to PartI (General Handout for all courses appended to the Timetable) this portion
gives further specific details reg
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Fall 2014
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCEPILANI, K. K. Birla
GOA CAMPUS
FIRST SEMESTER 20132014
Course Handout (Part II)
Date: 01/08/2013
In addition to partI (General Handout for all courses appended to the time table), this portion
gives further spec
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Fall 2014
BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI
K K BIRLA GOA CAMPUS
FIRST SEMESTER 201213
Course Handout (Part II)
Dated: 25/07/2013
In addition to partI, (General Handout for all courses appended to the time table) this
portion gives further specific
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
UNIT DT: Multiple Choice Questions
Lectures in Discrete Mathematics, Course 2, Bender/Williamson
Review Questions
Multiple Choice Questions for Review
1. In each case, two permutations on 6 are listed. In which case is the rst permutation
less than the se
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
UNIT Fn:Multiple Choice Questions
Lectures in Discrete Mathematics, Course 2, Bender/Williamson
Review Questions
Multiple Choice Questions for Review
1. In each case some information is given about a function. In which case is the information not sucient
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
UNIT EO: Multiple Choice Questions
Lectures in Discrete Mathematics, Course 1, Bender/Williamson
Equivalence and Order
Multiple Choice Questions for Review
In each case there is one correct answer (given at the end of the problem set). Try
to work the pro
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Arithmetic, Logic and Numbers
With and Introduction to Cryptography
Unit BF: Boolean Functions and Computer Arithmetic
Edward A. Bender
S. Gill Williamson
c Edward A. Bender & S. Gill Williamson 2010. All rights reserved.
Preface
The material in this unit
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Arithmetic, Logic and Numbers
With and Introduction to Cryptography
Unit Lo: Logic
Edward A. Bender
S. Gill Williamson
c Edward A. Bender & S. Gill Williamson 2010. All rights reserved.
Preface
The material in this unit of study was, over several years, p
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Solutions for Boolean Functions and Computer Arithmetic
BF1.1 The idea of this problem is to show how English phrases are translated into logical
expressions.
(a) f s. In English, but is and with an underlying message: The use of but
in this way often (b
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
UNIT BF: Multiple Choice Questions
Lectures in Discrete Mathematics, Course 1, Bender/Williamson
Review Questions
Multiple Choice Questions for Review
In each case there is one correct answer (given at the end of the problem set). Try
to work the problem
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Sets, Equivalence and Order
With an Introduction to Sequences and Series
Unit SF: Sets and Functions
Edward A. Bender
S. Gill Williamson
c Edward A. Bender & S. Gill Williamson 2010. All rights reserved.
Preface
The material in this unit of study was, ove
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
UNIT Lo: Multiple Choice Questions
Lectures in Discrete Mathematics, Course 1, Bender/Williamson,
Review Questions
Multiple Choice Questions for Review
In each case there is one correct answer (given at the end of the problem set). Try
to work the problem
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Arithmetic, Logic and Numbers
With an Introduction to Cryptography
Unit NT: Number Theory and Cryptography
Edward A. Bender
S. Gill Williamson
c Edward A. Bender & S. Gill Williamson 2010. All rights reserved.
Preface
The material in this unit of study wa
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Number Theory and Cryptography
Multiple Choice Questions for Review
UNIT NT: Multiple Choice Questions
Lectures in Discrete Mathematics,
Course 1, Bender/Williamson
In each case there is one correct answer (given at the end of the problem set). Try
to wor
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
UNIT SF: Multiple Choice Questions
Lectures in Discrete Mathematics, Course 1, Bender/Williamson
Review Questions
Multiple Choice Questions for Review
In each case there is one correct answer (given at the end of the problem set). Try
to work the problem
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Sets, Equivalence and Order
With an Introduction to Sequences and Series
Unit IS: Induction, Sequences and Series
Edward A. Bender
S. Gill Williamson
c Edward A. Bender & S. Gill Williamson 2010. All rights reserved.
Preface
The material in this unit of s
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Notation Index
(there exists)
(for all)
Fn4
Fn4
(such that) Fn4
Bn (Bell numbers) CL27
s t (equivalence relation) GT5
n
k
(binomial coecient) CL15
n
m1 ,m2 ,.
(multinomial
coecient) CL20
Probability notation
X (expectation, or
mean) Fn24
(X, Y
Birla Institute of Technology & Science, Pilani  Hyderabad
Discrete Mathematics
MATH 457

Spring 2013
Sets, Equivalence and Order
With an Introduction to Sequences and Series
Unit EO: Equivalence and Order
Edward A. Bender
S. Gill Williamson
c Edward A. Bender & S. Gill Williamson 2010. All rights reserved.
Preface
The material in this unit of study was,