CS154 Formal Languages and Computability Thaddeus Aid Department of Computer Science San Jose State University Spring 2016 Creative Commons Attribution-ShareAlike 4.0 International License

Creative Commons Attribution-ShareAlike 4.0 International License Citation Lecture developed in conjunction with: Introduction to Theory of Computation Anil Maheshwari Michiel Smid http://cglab.ca/~michiel/TheoryOfComputation/TheoryOfComputation.pdf Creative Commons Attribution-ShareAlike 4.0 International License

Creative Commons Attribution-ShareAlike 4.0 International License Theorem A mathematical statement that is true How do we know it is true? We prove it This is different to science where we test hypotheses Creative Commons Attribution-ShareAlike 4.0 International License

Creative Commons Attribution-ShareAlike 4.0 International License Proof Like debugging it is a bit of an art that takes time to learn Study and understand the problem Study and understand sub-theorems Construct a set of simple instance problems and solve them Write your steps down in order to remember what you did and spot errors! Keep at it! If it was easy then we wouldn’t need advanced mathematics! Creative Commons Attribution-ShareAlike 4.0 International License

Creative Commons Attribution-ShareAlike 4.0 International License Direct Proof Directly solve the proof Prove that for every positive odd integer n : n2 is also odd Prove that for every graph G = (V, E) the sum of all deg(v) is even Creative Commons Attribution-ShareAlike 4.0 International License

Creative Commons Attribution-ShareAlike 4.0 International License Constructive Proof Building the object, proves the object. Build an object such that the object has the property P A graph is called 3-regular if each deg(v) == 3 Prove that for any even n > 4 there exists a 3-regular graph Creative Commons Attribution-ShareAlike 4.0 International License

Nonconstructive Proof Determine if an object has the property P without constructing it Prove that there are irrational numbers x, y such that xy is rational Creative Commons Attribution-ShareAlike 4.0 International License

Proof by Contradiction If a theorem states that statement S is true try and prove the opposite Prove (using our constructive proof) that if n2 is even then n is even Prove sqrt(2) is irrational: sqrt(2) cannot be written as m/n where m and n are integers. Creative Commons Attribution-ShareAlike 4.0 International License

The Pigeon Hole Principle If n+1 objects are placed into n boxes then one box has at least 2 objects If |A| > |B| then there is no one to one mapping from A -> B Prove that for any positive integer n that n2 + 1 arbitrary positive numbers contains within it a subsequence of n + 1 numbers in either an ascending or descending order. Creative Commons Attribution-ShareAlike 4.0 International License

Creative Commons Attribution-ShareAlike 4.0 International License Proof by Induction Prove that step one is true: P(n) is true Induction step: P(n+1) is also true Prove: Creative Commons Attribution-ShareAlike 4.0 International License