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Lecture9_Induction

# Lecture9_Induction - Ling 726 Mathematical Linguistics...

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Ling 726: Mathematical Linguistics, Lecture 9: Induction. V. Borschev and B. Partee, October 19, 2004 p. 1 1 Lecture 9: Proof by Induction. 8.1.1. Recursive definitions. .................................................................................................................................... 1 8.4. Peano’s axioms and Proof by Induction. .......................................................................................................... 2 Homework 10 . ......................................................................................................................................................... 4 APPENDIX: . ........................................................................................................................................................... 5 How to use Induction in a Proof. ........................................................................................................................ 5 Introduction. ................................................................................................................................................... 5 Example 1 . ..................................................................................................................................................... 5 Example 2 . ..................................................................................................................................................... 7 Conclusion . .................................................................................................................................................... 8 Feedback. ....................................................................................................................................................... 8 Read: PtMW, Chapter 8, Section 8.4 (192-198) and Section 8.5.7 (214-215). Attention: Induction is a mind-bender, more than it seems at first! 8.1.1. Recursive definitions. First let’s review recursive definitions from Section 8.1.1 of PtMW. There is in fact a close connection between being able to specify the membership of some set recursively and being able to use some version of the Principle of Mathematical Induction to prove that all members of the set have some property or other. Consider the set M of all even-length mirror-image strings on { a,b }. An even-length mirror- image string is a string that can be divided into two halves, with the right half a mirror-image reversal of the left half (a “palindrome”). Examples: abba, babbab, aaaa, bbabbabb . Non- examples: babb, aaab, bab . Recursive definition of M : (8-1) 1. aa M & bb M 2. ( x )( x M ( axa M & bxb M )) 3. Nothing is in M except by virtue of rules 1 and 2. (Line 1 is called the

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Lecture9_Induction - Ling 726 Mathematical Linguistics...

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