Discrete And Combinatorial Mathematics What is Discrete Mathematics
Discrete And Combinatorial Mathematics • What is Discrete Mathematics? – “…the mathematics necessary for decision making in noncontinuous 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.
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.
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.
What is this?
What is this? This is not the United States. This is a model of the United States.
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: x. U => x. UI. 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: Mx. IIIy => Mx. Uy. d. ) Two U’s in a row can be removed from the word: Mx. UUy => 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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