Discrete And Combinatorial Mathematics

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Discrete And Combinatorial Mathematics

• What is Discrete Mathematics?
• the mathematics necessary for decision making
  in non-continuous situations. (ref
  http//math.about.com )
• Three main areas of Discrete Mathematics
• Existence Problems does a solution exist?
• Counting Problems how many are there?
• Optimization Problems What is the best solution
• Includes sets, functions, relations, matrix
  algebra, combinatorics, graph theory, logic,
  algorithmic thinking.

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Discrete And Combinatorial Mathematics

• What is Combinatorics?
• The branch of mathematics studying the
  enumeration, combination, and permutation of sets
  of elements and the mathematical relations which
  characterize these properties. (ref
  http//mathworld.wolfram.com )
• Includes sets, enumerations, combinations,
  permutations, and graph theory.

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Why are we Studying it?

• Real world systems are far too complex and
  ambiguous for us to reach any definitive
  conclusions about.
• Scientists, mathematicians, and engineers use
  models of the real world to predict real world
  results. The use of the model reduces ambiguity
  (and also reduces expressiveness).
• Formal models go one step further to remove all
  ambiguity.
• Start with a few given truths.
• Provide operations to combine them.
• Develop complex truths from the simple known
  ones.

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What is this?
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What is this?
This is not the United States. This is a model
of the United States.
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The MU-Puzzle
Hofstaders MU Puzzle (1) Given an alphabet
containing only M, I, and U, and the following
rules a.) An I can be placed at the end of a
word ending in U xU gt xUI . b.) The part of
the word after the first M can be repeated Mx gt
Mxx. c.) Three Is in a row can be replaced by a
single U MxIIIy gt MxUy. d.) Two Us in a row
can be removed from the word MxUUy gt Mxy. Where
x, y are strings of letters M, I and U. Start
with the word MI and you win when you create
the word MU. (1) From Godel, Escher, Bach
An Eternal Golden Braid by Douglas Hofstader.