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Transcript
JHU-CTY Theory of Computation (TCOM) Lancaster 2007 ~ Instructors Kayla Jacobs & Adam Groce REVIEW: MISCELLANEOUS MATH TOPICS Special Number Sets • Natural numbers: • Integers: = {1, 2, 3, 4, ...} = {..., -3, -2, -1, 0, 1, 2, 3, ...} • Rational numbers: • Real numbers: . Includes all rational numbers and irrational numbers (which can’t be rewritten as fractions but have decimal representations: e.g. π, e, and √2). Proof Techniques • Proof by Construction To show something can be done, make something (e.g. a TM) that can do it. • Proof by Contradiction Assume, for contradiction, the opposite of the statement you’re trying to prove. Then do stuff to reach a contradiction. Conclude that your assumption must be false after all. • Proof by Induction Base case: Prove the statement is true for n=1 Inductive hypothesis: Assume that the statement is true for n=m Inductive case: Prove the statement is true for n=m+1, using as a given that it’s true for n=m. • Pigeonhole Principle If you have n pigeons and m holes, and m≤n, then at least one hole has two or more pigeons. Combinatorics • Permutations: The number of ways to choose r from n elements, where order matters. nPr = n! / (n-r)! • Combinations: Number of ways to choose r from n elements, where order does not matter. nCr = n! / r!(n-r)!