Name of the Student:
|
|
- Abigail Morrison
- 5 years ago
- Views:
Transcription
1 SUBJECT NAME : Discrete Mathematics SUBJECT CODE : MA 65 MATERIAL NAME : Part A questios MATERIAL CODE : JM08AM1013 REGULATION : R008 UPDATED ON : May-Jue 016 (Sca the above Q.R code for the direct dowload of this material) Name of the Studet: Brach: Uit I (Logic ad Proofs) 1. Defie Tautology with a example.. Defie a rule of Uiversal specificatio. 3. Fid the truth table for the statemet P Q. 4. Costruct a truth table for the compoud propositio ( p q) ( q p) 5. Usig truth table show that P ( P Q) P.. 6. Costruct the truth table for the compoud propositio ( p q) ( p q) 7. Give P = {, 3,4,5,6}, state the truth value of the statemet ( x P )( x + 3 = 10) + =.. 8. Show that the propositios p qad p q are logically equivalet. 9. Show that ( p r ) ( q r ) ad ( ) 10. Prove that p, p q, q r r. p q r are logically equivalet. 11. Without usig truth table show that P ( Q P ) P ( P Q) ( P Q R ) (( P Q) ( P R) ) 1. Show that ( ). is a tautology. 13. Usig truth table, show that the propositio p ( p q) is a tautology. Prepared by C.Gaesa, M.Sc., M.Phil., (Ph: ) Page 1
2 ( ) p p q q a tautology? 14. Is ( ) 15. Write the egatio of the statemet( x)( y) p( x, y). 16. What are the egatios of the statemets x ( x > x) ad x ( x = ) =? 17. Give a idirect proof of the theorem If 3 + is odd, the is odd. 18. Whe do you say that two compoud propositios are equivalet? 19. What are the cotra positive, the coverse ad the iverse of the coditioal statemet If you work hard the you will be rewarded. 0. Symbolically express the followig statemet. It is ot true that 5 star ratig always meas good food ad good service. Uit II (Combiatorics) 1. State the priciple of strog iductio.. Use mathematical iductio to show that + 1!, = 1,, 3,... =. ( + 1) 3. Use mathematical iductio to show that =. 4. If seve colours are used to pait 50 bicycles, the show that at least 8 bicycles will be the same colour. 5. Write the geeratig fuctio for the sequece 6. Fid the recurrece relatio for the Fiboacci sequece , a, a, a, a, Fid the recurrece relatio satisfyig the equatio y = A(3) + B( 4). 8. Solve the recurrece relatio y ( k ) y ( k ) y ( k ) y () = 16ad y (3) = Solve: ak = 3ak 1, for k 1, with a 0 = = 0for k, where 10. Fid the umber of o-egative iteger solutios of the equatio x 1 + x + x 3 = State Pigeohole priciple. Prepared by C.Gaesa, M.Sc., M.Phil., (Ph: ) Page
3 1. What is well orderig priciple? 13. I how may ways ca all the letters i MATHEMATICAL be arraged. 14. What is the umber of arragemets of all the six letters i the word PEPPER? 15. How may permutatios of { a, b, c, d, e, f, g } ad with a? 16. How may differet bit strigs are there of legth seve? C, = C, Show that ( ) ( ) Uit III (Graph Theory) 1. Defie Pseudo graph.. Defie complete graph ad give a example. 3. Defie a regular graph. Ca a complete graph be a regular graph? 4. Whe is a simple graph G bipartite? Give a example. 5. Defie a coected graph ad a discoected graph with examples. 6. Defie strogly coected graph. 7. Is the directed graph give below strogly coected? Why or why ot? 8. Defie complete bipartite graph. 9. Draw a complete bipartite graph of K,3ad K 3, Defie isomorphism of two graphs. 11. Defie self complemetary graph. 1. Give a example of a Euler graph. 13. Give a example of a o-euleria graph which is Hamiltoia. Prepared by C.Gaesa, M.Sc., M.Phil., (Ph: ) Page 3
4 14. Give a example of a graph which is Euleria but ot Hamiltoia. 15. State the hadshakig theorem. 16. State the ecessary ad sufficiet coditios for the existece of a Euleria path i a coected graph. 17. Defie complemetary graph G of a simple graph G. If the degree sequece of the simple graph is 4, 3, 3,,, what is the degree sequece of G. 18. For which value of m ad does the complete bipartite graph K m, have a (i) Euler circuit (tour) (ii) Hamilto circuit (cycle). 19. Draw the graph represeted by the give adjacecy matrix Obtai the adjacecy matrix of the graph give below. Uit IV (Algebraic Structures) 1. Defie a semigroup.. Defie mooids. 3. State ay two properties of a group. 4. Prove that idetity elemet is uique i a group. 5. Whe is a group( G,*) called abelia? 6. Prove that every subgroup of a abelia group is ormal. 7. Prove that the idetity of a subgroup is the same as that of the group. Prepared by C.Gaesa, M.Sc., M.Phil., (Ph: ) Page 4
5 8. Defie homomorphism ad isomorphism betwee two algebraic systems. 9. Give a example for homomorphism. 10. If aad bare ay two elemets of a group G,*, show that Gis a abelia group if a * b = a * b. ad oly if ( ) 11. State Lagrage s theorem i group theory. 1. Show that every cyclic group is abelia. 13. Let M,*, e M be a mooidad a M. If aivertible, the show that its iverse is uique. 14. If ' a' is a geerator of a cyclic group G, the show that 15. Show that the set of all elemets aof a group (, ) x Gis a subgroup of G. 1 a is also a geerator of G. G such that a x = x afor every 16. Obtai all the distict left cosets of { [ 0 ],[ 3 ] } i the group ( 6, 6 ) uio. 17. Defie rig ad give a example. 18. Defie a field i a algebraic system. 19. Give a example of a rig which is ot a field. 0. Defie a commutative rig. Uit V (Lattices ad Boolea algebra) 1. Draw the Hasse diagram of, relatio be such that x X, where {,4,5,10,1, 0,5} y is xad y. Z + ad fid their X = ad the. Let X = { 1,, 3,4,6,8,1,4} ad R be a divisio relatio. Fid the Hasse diagram of the poset X, R. = ad ρ ( A) be its power set. Draw a Hasse diagram of ρ( A),. 3. Let A { a, b, c} 4. Show that least upper boud of a subset B i a poset ( A, ) is uique is it exists. 5. Defie a lattice. Give suitable example. Prepared by C.Gaesa, M.Sc., M.Phil., (Ph: ) Page 5
6 6. Defie sub-lattice. 7. Whe a lattice is called complete? 8. Whe is a lattice said to be bouded? 9. Defie lattice homomorphism. 10. Show that i a distributive lattice, if complemet of a elemet exits the it must be uique. 11. Give a example of a distributive lattice but ot complemeted. 1. I a Lattice ( L, ), prove that ( ) 13. Show that i a lattice if a b c, the a b = b c a a b = a, for all a, b L. = ad( a b) ( b c) = ( a b) ( b c) =. 14. Check whether the posets {( ) } Justify your claim. 15. Defie a Boolea algebra. { } 1, 3,6,9, D ad ( ) 16. Whe is a lattice said to be a Boolea algebra? 17. Is there a Boolea algebra with five elemets? Justify your aswer. 18. Prove the Boolea idetity: a. b + a. b = a. 19. Show that i a Boolea algebra ab + a b = 0if ad oly if a = b. 0. Show that the absorptio laws are valid i a Boolea algebra. 1,5, 5,15, D are lattices or ot All the Best---- Prepared by C.Gaesa, M.Sc., M.Phil., (Ph: ) Page 6
Name of the Student:
SUBJECT NAME : Discrete Mathematics SUBJECT CODE : MA 65 MATERIAL NAME : Problem Material MATERIAL CODE : JM08ADM010 (Sca the above QR code for the direct dowload of this material) Name of the Studet:
More informationUnit I (Logic and Proofs)
SUBJECT NAME SUBJECT CODE : MA 6566 MATERIAL NAME REGULATION : Discrete Mathematics : Part A questions : R2013 UPDATED ON : April-May 2018 (Scan the above QR code for the direct download of this material)
More informationModern Algebra. Previous year Questions from 2017 to Ramanasri
Moder Algebra Previous year Questios from 017 to 199 Ramaasri 017 S H O P NO- 4, 1 S T F L O O R, N E A R R A P I D F L O U R M I L L S, O L D R A J E N D E R N A G A R, N E W D E L H I. W E B S I T E
More informationREVIEW FOR CHAPTER 1
REVIEW FOR CHAPTER 1 A short summary: I this chapter you helped develop some basic coutig priciples. I particular, the uses of ordered pairs (The Product Priciple), fuctios, ad set partitios (The Sum Priciple)
More informationGENERALIZATIONS OF ZECKENDORFS THEOREM. TilVIOTHY J. KELLER Student, Harvey Mudd College, Claremont, California
GENERALIZATIONS OF ZECKENDORFS THEOREM TilVIOTHY J. KELLER Studet, Harvey Mudd College, Claremot, Califoria 91711 The Fiboacci umbers F are defied by the recurrece relatio Fi = F 2 = 1, F = F - + F 0 (
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationExercises 1 Sets and functions
Exercises 1 Sets ad fuctios HU Wei September 6, 018 1 Basics Set theory ca be made much more rigorous ad built upo a set of Axioms. But we will cover oly some heuristic ideas. For those iterested studets,
More informationTutorial F n F n 1
(CS 207) Discrete Structures July 30, 203 Tutorial. Prove the followig properties of Fiboacci umbers usig iductio, where Fiboacci umbers are defied as follows: F 0 =0,F =adf = F + F 2. (a) Prove that P
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More informationName of the Student:
SUBJECT NAME : Trasforms ad Partial Diff Eq SUBJECT CODE : MA MATERIAL NAME : Problem Material MATERIAL CODE : JM8AM6 REGULATION : R8 UPDATED ON : April-May 4 (Sca the above QR code for the direct dowload
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
More information# fixed points of g. Tree to string. Repeatedly select the leaf with the smallest label, write down the label of its neighbour and remove the leaf.
Combiatorics Graph Theory Coutig labelled ad ulabelled graphs There are 2 ( 2) labelled graphs of order. The ulabelled graphs of order correspod to orbits of the actio of S o the set of labelled graphs.
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationLecture Notes for CS 313H, Fall 2011
Lecture Notes for CS 313H, Fall 011 August 5. We start by examiig triagular umbers: T () = 1 + + + ( = 0, 1,,...). Triagular umbers ca be also defied recursively: T (0) = 0, T ( + 1) = T () + + 1, or usig
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More information2.4 - Sequences and Series
2.4 - Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2,
More informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
More informationDifferent kinds of Mathematical Induction
Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}
More informationA brief introduction to linear algebra
CHAPTER 6 A brief itroductio to liear algebra 1. Vector spaces ad liear maps I what follows, fix K 2{Q, R, C}. More geerally, K ca be ay field. 1.1. Vector spaces. Motivated by our ituitio of addig ad
More informationCOS 341 Discrete Mathematics. Exponential Generating Functions and Recurrence Relations
COS 341 Discrete Mathematics Epoetial Geeratig Fuctios ad Recurrece Relatios 1 Tetbook? 1 studets said they eeded the tetbook, but oly oe studet bought the tet from Triagle. If you do t have the book,
More information} is said to be a Cauchy sequence provided the following condition is true.
Math 4200, Fial Exam Review I. Itroductio to Proofs 1. Prove the Pythagorea theorem. 2. Show that 43 is a irratioal umber. II. Itroductio to Logic 1. Costruct a truth table for the statemet ( p ad ~ r
More information24 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS
24 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS Corollary 2.30. Suppose that the semisimple decompositio of the G- module V is V = i S i. The i = χ V,χ i Proof. Sice χ V W = χ V + χ W, we have:
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationHomework 3. = k 1. Let S be a set of n elements, and let a, b, c be distinct elements of S. The number of k-subsets of S is
Homewor 3 Chapter 5 pp53: 3 40 45 Chapter 6 p85: 4 6 4 30 Use combiatorial reasoig to prove the idetity 3 3 Proof Let S be a set of elemets ad let a b c be distict elemets of S The umber of -subsets of
More informationLecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =
COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.
More informationSection 4.3. Boolean functions
Sectio 4.3. Boolea fuctios Let us take aother look at the simplest o-trivial Boolea algebra, ({0}), the power-set algebra based o a oe-elemet set, chose here as {0}. This has two elemets, the empty set,
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationLecture 10: Mathematical Preliminaries
Lecture : Mathematical Prelimiaries Obective: Reviewig mathematical cocepts ad tools that are frequetly used i the aalysis of algorithms. Lecture # Slide # I this
More informationHomework 2 January 19, 2006 Math 522. Direction: This homework is due on January 26, In order to receive full credit, answer
Homework 2 Jauary 9, 26 Math 522 Directio: This homework is due o Jauary 26, 26. I order to receive full credit, aswer each problem completely ad must show all work.. What is the set of the uits (that
More informationMA541 : Real Analysis. Tutorial and Practice Problems - 1 Hints and Solutions
MA54 : Real Aalysis Tutorial ad Practice Problems - Hits ad Solutios. Suppose that S is a oempty subset of real umbers that is bouded (i.e. bouded above as well as below). Prove that if S sup S. What ca
More informationCSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo
Coutig Methods CSE 191, Class Note 05: Coutig Methods Computer Sci & Eg Dept SUNY Buffalo c Xi He (Uiversity at Buffalo CSE 191 Discrete Structures 1 / 48 Need for Coutig The problem of coutig the umber
More informationSets. Sets. Operations on Sets Laws of Algebra of Sets Cardinal Number of a Finite and Infinite Set. Representation of Sets Power Set Venn Diagram
Sets MILESTONE Sets Represetatio of Sets Power Set Ve Diagram Operatios o Sets Laws of lgebra of Sets ardial Number of a Fiite ad Ifiite Set I Mathematical laguage all livig ad o-livig thigs i uiverse
More informationA B = φ No conclusion. 2. (5) List the values of the sets below. Let A = {n 2 : n P n 5} = {1,4,9,16,25} and B = {n 4 : n P n 5} = {1,16,81,256,625}
CPSC 070 Aswer Keys Test # October 1, 014 1. (a) (5) Defie (A B) to be those elemets i set A but ot i set B. Use set membership tables to determie what elemets are cotaied i (A (B A)). Use set membership
More informationEnd-of-Year Contest. ERHS Math Club. May 5, 2009
Ed-of-Year Cotest ERHS Math Club May 5, 009 Problem 1: There are 9 cois. Oe is fake ad weighs a little less tha the others. Fid the fake coi by weighigs. Solutio: Separate the 9 cois ito 3 groups (A, B,
More informationMath 475, Problem Set #12: Answers
Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe
More informationChapter 0. Review of set theory. 0.1 Sets
Chapter 0 Review of set theory Set theory plays a cetral role i the theory of probability. Thus, we will ope this course with a quick review of those otios of set theory which will be used repeatedly.
More informationInjections, Surjections, and the Pigeonhole Principle
Ijectios, Surjectios, ad the Pigeohole Priciple 1 (10 poits Here we will come up with a sloppy boud o the umber of parethesisestigs (a (5 poits Describe a ijectio from the set of possible ways to est pairs
More informationMath 4107: Abstract Algebra I Fall Webwork Assignment1-Groups (5 parts/problems) Solutions are on Webwork.
Math 4107: Abstract Algebra I Fall 2017 Assigmet 1 Solutios 1. Webwork Assigmet1-Groups 5 parts/problems) Solutios are o Webwork. 2. Webwork Assigmet1-Subgroups 5 parts/problems) Solutios are o Webwork.
More informationResolution Proofs of Generalized Pigeonhole Principles
Resolutio Proofs of Geeralized Pigeohole Priciples Samuel R. Buss Departmet of Mathematics Uiversity of Califoria, Berkeley Győrgy Turá Departmet of Mathematics, Statistics, ad Computer Sciece Uiversity
More informationPROBLEMS ON ABSTRACT ALGEBRA
PROBLEMS ON ABSTRACT ALGEBRA 1 (Putam 197 A). Let S be a set ad let be a biary operatio o S satisfyig the laws x (x y) = y for all x, y i S, (y x) x = y for all x, y i S. Show that is commutative but ot
More informationDisjoint unions of complete graphs characterized by their Laplacian spectrum
Electroic Joural of Liear Algebra Volume 18 Volume 18 (009) Article 56 009 Disjoit uios of complete graphs characterized by their Laplacia spectrum Romai Boulet boulet@uiv-tlse.fr Follow this ad additioal
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More information(k) x n. n! tk = x a. (i) x p p! ti ) ( q 0. i 0. k A (i) n p
Math 880 Bigraded Classes & Stirlig Cycle Numbers Fall 206 Bigraded classes. Followig Flajolet-Sedgewic Ch. III, we defie a bigraded class A to be a set of combiatorial objects a A with two measures of
More information10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term.
0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece
More information1 Counting and Stirling Numbers
1 Coutig ad Stirlig Numbers Natural Numbers: We let N {0, 1, 2,...} deote the set of atural umbers. []: For N we let [] {1, 2,..., }. Sym: For a set X we let Sym(X) deote the set of bijectios from X to
More informationChapter 2. Periodic points of toral. automorphisms. 2.1 General introduction
Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad
More informationSection 5.1 The Basics of Counting
1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of
More informationLECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK)
LECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK) Everythig marked by is ot required by the course syllabus I this lecture, all vector spaces is over the real umber R. All vectors i R is viewed as a colum
More informationOn Net-Regular Signed Graphs
Iteratioal J.Math. Combi. Vol.1(2016), 57-64 O Net-Regular Siged Graphs Nuta G.Nayak Departmet of Mathematics ad Statistics S. S. Dempo College of Commerce ad Ecoomics, Goa, Idia E-mail: ayakuta@yahoo.com
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More information3.1 Counting Principles
3.1 Coutig Priciples Goal: Cout the umber of objects i a set. Notatio: Whe S is a set, S deotes the umber of objects i the set. This is also called S s cardiality. Additio Priciple: Whe you wat to cout
More informationBooks Recommended for Further Reading
Books Recommeded for Further Readig by 8.5..8 o 0//8. For persoal use oly.. K. P. Bogart, Itroductory Combiatorics rd ed., S. I. Harcourt Brace College Publishers, 998.. R. A. Brualdi, Itroductory Combiatorics
More informationPairs of disjoint q-element subsets far from each other
Pairs of disjoit q-elemet subsets far from each other Hikoe Eomoto Departmet of Mathematics, Keio Uiversity 3-14-1 Hiyoshi, Kohoku-Ku, Yokohama, 223 Japa, eomoto@math.keio.ac.jp Gyula O.H. Katoa Alfréd
More informationReal Variables II Homework Set #5
Real Variables II Homework Set #5 Name: Due Friday /0 by 4pm (at GOS-4) Istructios: () Attach this page to the frot of your homework assigmet you tur i (or write each problem before your solutio). () Please
More information2.4 Sequences, Sequences of Sets
72 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.4 Sequeces, Sequeces of Sets 2.4.1 Sequeces Defiitio 2.4.1 (sequece Let S R. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For each
More informationMathematics Extension 1
016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved
More informationMATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006
MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the
More informationThe Boolean Ring of Intervals
MATH 532 Lebesgue Measure Dr. Neal, WKU We ow shall apply the results obtaied about outer measure to the legth measure o the real lie. Throughout, our space X will be the set of real umbers R. Whe ecessary,
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationM A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O
M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #4 CORRECTION Algebra I 1 4 / 1 0 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O P r o f e s s o r : D r o r B a r - N a t a Correctio Homework Assigmet
More informationTeacher s Marking. Guide/Answers
WOLLONGONG COLLEGE AUSRALIA eacher s Markig A College of the Uiversity of Wollogog Guide/Aswers Diploa i Iforatio echology Fial Exaiatio Autu 008 WUC Discrete Matheatics his exa represets 60% of the total
More informationWhat is Probability?
Quatificatio of ucertaity. What is Probability? Mathematical model for thigs that occur radomly. Radom ot haphazard, do t kow what will happe o ay oe experimet, but has a log ru order. The cocept of probability
More informationProblem. Consider the sequence a j for j N defined by the recurrence a j+1 = 2a j + j for j > 0
GENERATING FUNCTIONS Give a ifiite sequece a 0,a,a,, its ordiary geeratig fuctio is A : a Geeratig fuctios are ofte useful for fidig a closed forula for the eleets of a sequece, fidig a recurrece forula,
More informationACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory
1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.
More informationNBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian?
NBHM QUESTION 7 NBHM QUESTION 7 NBHM QUESTION 7 Sectio : Algebra Q Let G be a group of order Which of the followig coditios imply that G is abelia? 5 36 Q Which of the followig subgroups are ecesarily
More information1 Introduction. 1.1 Notation and Terminology
1 Itroductio You have already leared some cocepts of calculus such as limit of a sequece, limit, cotiuity, derivative, ad itegral of a fuctio etc. Real Aalysis studies them more rigorously usig a laguage
More informationDiscrete mathematics , Fall Instructor: prof. János Pach. 1 Counting problems and the inclusion-exclusion principle
Discrete mathematics 014-015, Fall Istructor: prof Jáos Pach - covered material - Special thaks to Jaa Cslovjecsek, Kelly Fakhauser, ad Sloboda Krstic for sharig their lecture otes If you otice ay errors,
More information1 Summary: Binary and Logic
1 Summary: Biary ad Logic Biary Usiged Represetatio : each 1-bit is a power of two, the right-most is for 2 0 : 0110101 2 = 2 5 + 2 4 + 2 2 + 2 0 = 32 + 16 + 4 + 1 = 53 10 Usiged Rage o bits is [0...2
More informationRegn. No. North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: ,
. Sectio-A cotais 30 Multiple Choice Questios (MCQ). Each questio has 4 choices (a), (b), (c) ad (d), for its aswer, out of which ONLY ONE is correct. From Q. to Q.0 carries Marks ad Q. to Q.30 carries
More informationOptimization Methods MIT 2.098/6.255/ Final exam
Optimizatio Methods MIT 2.098/6.255/15.093 Fial exam Date Give: December 19th, 2006 P1. [30 pts] Classify the followig statemets as true or false. All aswers must be well-justified, either through a short
More informationChapter 1 : Combinatorial Analysis
STAT/MATH 394 A - PROBABILITY I UW Autum Quarter 205 Néhémy Lim Chapter : Combiatorial Aalysis A major brach of combiatorial aalysis called eumerative combiatorics cosists of studyig methods for coutig
More informationCATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01
CATHOLIC JUNIOR COLLEGE Geeral Certificate of Educatio Advaced Level Higher JC Prelimiary Examiatio MATHEMATICS 9740/0 Paper 4 Aug 06 hours Additioal Materials: List of Formulae (MF5) Name: Class: READ
More informationCSI 2101 Discrete Structures Winter Homework Assignment #4 (100 points, weight 5%) Due: Thursday, April 5, at 1:00pm (in lecture)
CSI 101 Discrete Structures Witer 01 Prof. Lucia Moura Uiversity of Ottawa Homework Assigmet #4 (100 poits, weight %) Due: Thursday, April, at 1:00pm (i lecture) Program verificatio, Recurrece Relatios
More informationAH Checklist (Unit 3) AH Checklist (Unit 3) Matrices
AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow
More informationFind a formula for the exponential function whose graph is given , 1 2,16 1, 6
Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is
More informationPermutations & Combinations. Dr Patrick Chan. Multiplication / Addition Principle Inclusion-Exclusion Principle Permutation / Combination
Discrete Mathematic Chapter 3: C outig 3. The Basics of Coutig 3.3 Permutatios & Combiatios 3.5 Geeralized Permutatios & Combiatios 3.6 Geeratig Permutatios & Combiatios Dr Patrick Cha School of Computer
More informationFermat s Little Theorem. mod 13 = 0, = }{{} mod 13 = 0. = a a a }{{} mod 13 = a 12 mod 13 = 1, mod 13 = a 13 mod 13 = a.
Departmet of Mathematical Scieces Istructor: Daiva Puciskaite Discrete Mathematics Fermat s Little Theorem 43.. For all a Z 3, calculate a 2 ad a 3. Case a = 0. 0 0 2-times Case a 0. 0 0 3-times a a 2-times
More informationStructural Functionality as a Fundamental Property of Boolean Algebra and Base for Its Real-Valued Realizations
Structural Fuctioality as a Fudametal Property of Boolea Algebra ad Base for Its Real-Valued Realizatios Draga G. Radojević Uiversity of Belgrade, Istitute Mihajlo Pupi, Belgrade draga.radojevic@pupi.rs
More informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
More informationDavenport-Schinzel Sequences and their Geometric Applications
Advaced Computatioal Geometry Sprig 2004 Daveport-Schizel Sequeces ad their Geometric Applicatios Prof. Joseph Mitchell Scribe: Mohit Gupta 1 Overview I this lecture, we itroduce the cocept of Daveport-Schizel
More informationLONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES
J Lodo Math Soc (2 50, (1994, 465 476 LONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES Jerzy Wojciechowski Abstract I [5] Abbott ad Katchalski ask if there exists a costat c >
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationA COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP
A COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP JEREMY BRAZAS AND LUIS MATOS Abstract. Traditioal examples of spaces that have ucoutable fudametal group (such as the Hawaiia earrig space) are path-coected
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationGenerating Functions. 1 Operations on generating functions
Geeratig Fuctios The geeratig fuctio for a sequece a 0, a,..., a,... is defied to be the power series fx a x. 0 We say that a 0, a,... is the sequece geerated by fx ad a is the coefficiet of x. Example
More informationMatrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.
2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a
More information3sin A 1 2sin B. 3π x is a solution. 1. If A and B are acute positive angles satisfying the equation 3sin A 2sin B 1 and 3sin 2A 2sin 2B 0, then A 2B
1. If A ad B are acute positive agles satisfyig the equatio 3si A si B 1 ad 3si A si B 0, the A B (a) (b) (c) (d) 6. 3 si A + si B = 1 3si A 1 si B 3 si A = cosb Also 3 si A si B = 0 si B = 3 si A Now,
More informationLecture 4: Grassmannians, Finite and Affine Morphisms
18.725 Algebraic Geometry I Lecture 4 Lecture 4: Grassmaias, Fiite ad Affie Morphisms Remarks o last time 1. Last time, we proved the Noether ormalizatio lemma: If A is a fiitely geerated k-algebra, the,
More informationHomework 9. (n + 1)! = 1 1
. Chapter : Questio 8 If N, the Homewor 9 Proof. We will prove this by usig iductio o. 2! + 2 3! + 3 4! + + +! +!. Base step: Whe the left had side is. Whe the right had side is 2! 2 +! 2 which proves
More informationInternational Baccalaureate LECTURE NOTES MATHEMATICS HL FURTHER MATHEMATICS HL Christos Nikolaidis TOPIC NUMBER THEORY
Iteratioal Baccalaureate LECTURE NOTES MATHEMATICS HL FURTHER MATHEMATICS HL TOPIC NUMBER THEORY METHODS OF PROOF. Couterexample - Cotradictio - Pigeohole Priciple Strog mathematical iductio 2 DIVISIBILITY....
More informationBasic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.
Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the
More informationREGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS
REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x]
More informationON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS
ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary
More information: Transforms and Partial Differential Equations
Trasforms ad Partial Differetial Equatios 018 SUBJECT NAME : Trasforms ad Partial Differetial Equatios SUBJECT CODE : MA 6351 MATERIAL NAME : Part A questios REGULATION : R013 WEBSITE : wwwharigaeshcom
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationRoger Apéry's proof that zeta(3) is irrational
Cliff Bott cliffbott@hotmail.com 11 October 2011 Roger Apéry's proof that zeta(3) is irratioal Roger Apéry developed a method for searchig for cotiued fractio represetatios of umbers that have a form such
More informationSequence A sequence is a function whose domain of definition is the set of natural numbers.
Chapter Sequeces Course Title: Real Aalysis Course Code: MTH3 Course istructor: Dr Atiq ur Rehma Class: MSc-I Course URL: wwwmathcityorg/atiq/fa8-mth3 Sequeces form a importat compoet of Mathematical Aalysis
More informationMath 5705: Enumerative Combinatorics, Fall 2018: Homework 3
Uiversity of Miesota, School of Mathematics Math 5705: Eumerative Combiatorics, all 2018: Homewor 3 Darij Griberg October 15, 2018 due date: Wedesday, 10 October 2018 at the begiig of class, or before
More information