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Discrete And Combinatorial Mathematics

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?

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Discrete And Combinatorial Mathematics

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  1. 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.

  2. 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.

  3. 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.

  4. What is this?

  5. What is this? This is not the United States. This is a model of the United States.

  6. The MU-Puzzle Hofstader’s 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 => xUI . b.) The part of the word after the first M can be repeated: Mx => Mxx. c.) Three I’s in a row can be replaced by a single U: MxIIIy => MxUy. d.) Two U’s in a row can be removed from the word: MxUUy => 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.

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