Math 245 - Writing Assignment

Writing  Assignments/Technology  Assignments     You  will  complete  a  total  of  six  writing/technology  assignments  throughout  the  semester.  You  can  choose   these  from  the  list  below.  They  can  all  be  writing  assignments,  or  they  could  all  be  technology   assignments  or  some  can  be  writing  and  some  can  be  technology.  The  dues  dates  are  listed  in  the   syllabus.    

Writing  Assignments  Requirements  

Each  writing  assignment  is  worth  20  points  should  include  the  following  sections:     Background  (3  points):  This  is  a  discussion  of  how  the  non-­‐mathematical  and  mathematical  portions  of  your  topic  fit   together.  You  might  include  a  historical  background  of  the  topic,  definitions  of  terms,  the  discrete  mathematics  ideas   that  are  addressed  (e.g.  induction,  logical  fallacy,  etc.),  and  some  explanation  about  why  these  ideas  were  useful.  

Examples  (10  points):  In  most  of  your  writing  assignments  you  are  asked  to  discuss  and  describe  an  aspect  of  discrete   mathematics.  Give  two  of  three  examples  or  techniques  of  the  topic  under  discussion.  Give  general  information  and  also   specific  examples  of  the  topic.  

Bibliography  (2  points):  List  the  references  you  used  to  complete  this  report.  Just  list  title  and  author  for  any  books  and   articles  you  used.  You  should  also  include  a  list  of  people  that  you  consulted  or  any  other  form  of  help  that  you  received.   For  example,  you  might  obtain  some  of  your  information  from  the  internet;  in  this  case,  you  could  include  the   website.  You'll  need  at  least  one  book  or  article  as  a  reference,  preferably  two,  and  a  total  of  at  least  two  references.  

You'll  notice  that  there  are  still  5  points  unaccounted  for.  The  remaining  5  points  are  for  style:  clarity,  neatness,  flow,   design,  organization  and  creativity-­‐-­‐it's  important  to  be  able  to  communicate  your  ideas.  

Note:  you  don't  have  to  put  your  report  in  the  precise  order  given  above.  You  may  prefer  to  use  the  assigned  problems   to  illustrate  how  the  ideas  of  the  subject  fit  together  with  the  mathematical  ideas  that  you  will  be  using,  in  which  case   Background  and  Examples  would  be  interwoven.  Just  make  sure  that  these  aspects  appear  in  your  report.  

 

Technology  Assignments  Requirements  

Each  technology  assignment  is  worth  20  points  should  include  the  following  sections:     Background  (5  points):  This  is  a  written  paragraph  of  how  the  non-­‐mathematical  and  mathematical  portions  of  your   topic  fit  together.  In  other  words,  you  need  to  talk  about  what  you  needed  to  know  about  your  topic  in  order  to  solve   the  problems  and  how  discrete  mathematics  fits  into  the  picture.  So  you  might  include  the  definitions  of  terms,  the   discrete  mathematics  ideas  you  used  (e.g.  induction,  logical  fallacy,  etc.),  and  some  explanation  about  why  these  ideas   were  useful.  

Solution  (15  points):  In  most  of  your  technology  assignments  you  are  asked  to  write  a  program  to  solve  a  problem.  Copy   and  paste  your  code  in  a  Word  document  and  annotate  each  section  of  the  code  with  explanation  of  what  are  you  doing   in  each  section.  Include  a  screen  shot  of  the  output  of  the  program.  If  you  are  using  EXCEL,  include  the  excel  file  as  a   separate  attachment.  

 

Writing  and  Technology  assignments  will  be  submitted  in  Canvas  and  checked  with  SafeAssign.  Make  sure  to  submit   your  own  work  and  give  written  explanation  using  your  own  voice.  Compile  your  work  in  one  document  and  save  it  in   pdf  format  and  submit  it  by  clicking  on  the  assignment  title.  

Writing  Assignments:   Combinatorics:   1.  Describe  some  of  the  earliest  uses  of  the  pigeonhole  principle  by  Dirichlet  and  other  mathematicians     2.  Describe  the  different  models  used  to  model  the  distribution  of  particles  in  statistical  mechanics,  including   Maxwell–Boltzmann,  Bose–Einstein,  and  Fermi–Dirac  statistics.  In  each  case,  describe  the  counting  techniques   used  in  the  model.     Logic:     3.  Discuss  logical  paradoxes,  including  the  paradox  of  Epimenides  the  Cretan,  Jourdain’s  card  paradox,  and  the   barber  paradox,  and  how  they  are  resolved.     4.  Describe  how  fuzzy  logic  is  being  applied  to  practical  applications.         Proofs  and  Induction:   5.  Look  up  some  of  the  incorrect  proofs  of  famous  open  questions  and  open  questions  that  were  solved  since  1970   and  describe  the  type  of  error  made  in  each  proof.   6.  Describe  the  origins  of  mathematical  induction.  Who  were  the  first  people  to  use  it  and  to  which  problems  did   they  apply  it?     Algorithms:   7.  Describe  six  different  NP-­‐complete  problems.     8.  Describe  the  historic  trends  in  how  quickly  processors  can  perform  operations  and  use  these  trends  to  estimate   how  quickly  processors  will  be  able  to  perform  operations  in  the  next  twenty  years.   Relations  and  Functions:   9.  Discuss  the  concept  of  a  fuzzy  relation.  How  are  fuzzy  relations  used?     10.  Describe  how  equivalence  classes  can  be  used  to  define  the  rational  numbers  as  classes  of  pairs  of  integers  and   how  the  basic  arithmetic  operations  on  rational  numbers  can  be  defined  following  this  approach.     Recursion:     11.  Describe  a  variety  of  different  applications  of  the  Fibonacci  numbers  to  the  biological  and  the  physical  sciences.     12.  When  are  the  numbers  of  a  sequence  truly  random  numbers,  and  not  pseudorandom?  What  shortcomings  have   been  observed  in  simulations  and  experiments  in  which  pseudorandom  numbers  have  been  used?  What  are  the   properties  that  pseudorandom  numbers  can  have  that  random  numbers  should  not  have?   Number  Theory:   13.  Describe  the  history  of  the  Chinese  remainder  theorem.  Describe  some  of  the  relevant  problems  posed  in   Chinese  and  Hindu  writings  and  how  the  Chinese  remainder  theorem  applies  to  them.     14.  Show  how  a  congruence  can  be  used  to  tell  the  day  of  the  week  for  any  given  date.     Graph  Theory:   15.  Discuss  the  applications  of  graph  theory  to  the  study  of  ecosystems,  to  sociology  and  to  psychology.     16.  Explain  how  graph  theory  can  help  uncover  networks  of  criminals  or  terrorists  by  studying  relevant  social  and   communication  networks.          

Programming  Assignments:   Sets:   17.  a)  Given  two  finite  sets,  list  all  elements  in  the  Cartesian  product  of  these  two  sets.  b)  Given  a  finite  set,  list  all   elements  of  its  power  set.     Combinatorics:   18.  Given  an  equation  𝑥! + 𝑥! + ⋯+ 𝑥! = 𝐶,  where  C  is  a  constant,  and  𝑥!,𝑥!,… ,𝑥!  are  nonnegative  integers,  list   all  the  solutions.     19.  Input  the  English  alphabet  (a  string  of  26  letters):  

a) Generate  all  the  permutations  of  a  set  with  four  elements   b) Generate  all  the  combinations  of  a  set  with  four  elements  

  Logic:     20.  Given  the  truth  values  of  the  propositions  p  and  q,  find  he  truth  values  of  the  conjunction,  disjunction,  exclusive   or,  conditional  statement,  and  biconditional  statement  of  these  prepositions.     21.  Find  as  many  positive  integers  as  you  can  that  can  be  written  as  the  sum  of  cubes  of  positive  integers,  in  two   different  ways,  sharing  this  property  with  the  number  1729     Algorithms:   22.  Given  an  ordered  list  of  n  distinct  integer,  determine  the  position  of  a  specific  integer  on  the  list  using:  

a) A  linear  search  algorithm   b) A  binary  search  algorithm   c) A  tertiary  search  algorithm  

  23.  Given  a  list  on  n  integers,  use  the  greedy  algorithm  to  find  the  change  for  n  cents  using  quarters,  dimes,  nickels   and  pennies.       Relations  and  Functions:   24.  Display  all  the  different  reflexive,  symmetric  and  transitive  relations  on  a  set  with  six  elements.   Recursion:     25.  Given  a  nonnegative  integer  n,  find  the  nth  Fibonacci  number  using  recursion.     26.  Determine  which  Fibonacci  numbers  are  divisible  by  5,  which  are  divisible  by  7,  and  which  are  divisible  by  11.   Prove  that  your  conjectures  are  correct.     Number  Theory:   27.  Given  integers  n  and  b,  each  greater  than  1,  find  the  base  b  expansion  of  this  integer.     28.  Given  a  positive  integer,  determine  whether  it  is  a  prime  number  or  a  composite  number  using  trial  division.  If   the  number  is  composite,  find  the  prime  factorization  of  the  number.       29.  Given  two  positive  integers,  find  their  least  common  multiple.     Graph  Theory:   30.  Given  the  vertex  pairs  associated  to  the  edges  of  a  graph,  construct  an  adjacency  matrix  for  the  graph.  (Produce   a  version  that  works  when  loops,  multiple  edges,  or  directed  edges  are  present.)     31.  Given  the  vertex  pairs  associated  to  the  edges  of  a  multigraph,  determine  whether  it  has  an  Euler  circuit  and,  if   not,  whether  it  has  an  Euler  path.  Construct  an  Euler  path  or  circuit  if  it  exists.     32.  Given  the  list  of  edges  and  weights  of  these  edges  of  a  weighted  connected  simple  graph  and  two  vertices  in  this   graph,  find  the  length  of  a  shortest  path  between  them  using  Dijkstra’s  algorithm.  Also,  find  a  shortest  path.