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# Theory of Computation: Lecture 03

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2. Proof by Induction

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### Transcript of "Theory of Computation: Lecture 03"

1. 1. CS 5000: Theory of Computation Lecture 03 Vladimir Kulyukin Department of Computer Science Utah State Universitywww.youtube.com/vkedco www.vkedco.blogspot.com
2. 2. Outline ● Two Proof Techniques Useful in CS – Contradiction – Inductionwww.youtube.com/vkedco www.vkedco.blogspot.com
3. 3. Review: Proof Techniques ● Proof techniques are independent of their subject matter: valid proofs in calculus use the same proof techniques as valid proofs in algorithms or theory of computation ● Common proof techniques can be identified ● Learning to identify common proof techniques will enable you to study many areas of CS independently ● The ability to identify proof techniques is based on your ability to understand how the technique works and when it is likely to be applicablewww.youtube.com/vkedco www.vkedco.blogspot.com
4. 4. Two Proof Techniques for CS ● There are two proof techniques that are prominent in CS: contradiction and induction ● The contradiction method is used any time you want to show that something is impossible ● The gist of the contradiction method is this: you assume that it is possible and then derive a statement that contradicts another statement that you and everyone else familiar with the subject matter know to be truewww.youtube.com/vkedco
5. 5. Two Proof Techniques for CS ● You use induction method to prove that some statement S is true for every natural number greater than or equal to some initial value k ● In other words, S(k) is true, S(k+1) is true, S(k+2) is true, S(k+3) is true, and so on to infinity ● In the end, a valid inductive proof allows you to conclude that S(n) is true for all numbers greater than or equal to kwww.youtube.com/vkedco www.vkedco.blogspot.com