1. CS154 Formal Languages and Computability
Thaddeus Aid
Department of Computer Science
San Jose State University
Spring 2016
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Citation
Lecture developed in conjunction with:
Introduction to Theory of Computation
Anil Maheshwari
Michiel Smid
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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
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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!
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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
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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
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7. 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
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8. 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.
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9. 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.
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Proof by Induction
Prove that step one is true: P(n) is true
Induction step: P(n+1) is also true
Prove:
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