09ma6cHW7

# 09ma6cHW7 - Ma/CS 6c Assignment#7 Due Thursday May 27 at 1...

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Unformatted text preview: Ma/CS 6c Assignment #7 Due Thursday, May 27 at 1 p.m. (20%) 1. Call a set X ⊆ N eventually periodic if there is n ∈ N and p ∈ N ,p ≥ 1 (a period ) such that for all n ≥ n , n ∈ X iff n + p ∈ X. Show that every eventually periodic set is first-order definable in the structure N = h N ; 0 ,S, + i . (It turns out that these are the only first-order definable sets in N .) We of course identify here subsets of N with unary relations on N . The language here is L = { 0; S, + } . (20%) 2. Show that + is not first-order definable in the structure N 00 = h N , ·i . The language here is L = {·} . (20%) 3. * Consider the language L = { f,g } with f,g unary function symbols, and its structure M = h Z × Z ,f M ,g M i , where f M ( i,j ) = ( i,j + 1) g M ( i,j ) = ( i + 1 ,j ) . Determine all the first-order definable subsets (i.e., unary relations) of M = Z × Z . (20%) 4. Consider the language L = { < } and it structure A = h R ,< i ....
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## This note was uploaded on 07/19/2010 for the course MA 6 taught by Professor Inessaepstein during the Spring '09 term at Caltech.

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09ma6cHW7 - Ma/CS 6c Assignment#7 Due Thursday May 27 at 1...

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