09ma6cHW6

09ma6cHW6 - A 1 ∀ x ∃ yP x,y A 2 ∀ x ∀ y ∀ z P...

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Ma/CS 6c Assignment #6 Due Thursday, May 13 at 1 p.m. (20%) 1. Prove that a proper initial segment of a formula in first-order logic is not a formula. (10%) 2. (a) Write a sentence A 6 in the language with no non-logical symbols (but recall that “=” is available), such that for any structure M = h M i for this language: M | = A 6 iff M has exactly 6 elements. (b) Write a sentence B in the language L = { R } , R a binary relation symbol, such that for any structure M = h M,R M i for this language, we have: M | = B iff R M is the graph of a one-to-one correspondence of M with itself (i.e., a permutation of M ) . (10%) 3. Find a sentence A in the language L = { < } , < a binary relation symbol, such that h Q ,< i | = A but h Z ,< i | = ¬ A. (Here Q = rationals, Z = integers, and < is the usual ordering in each one of them.) (10%) 4. Write a sentence A in the language L = { R } , R a binary relation symbol, such that for any structure M = h M,R M i for L we have: M | = A iff R m is an equivalence relation on M which has exactly 4 equivalence classes each of which has exactly 8 elements. (20%) 5 * . Consider the language L = { P } , P a binary relation symbol, and the sentences
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Unformatted text preview: A 1 : ∀ x ∃ yP ( x,y ) A 2 : ∀ x ∀ y ∀ z [ P ( x,y ) ∧ P ( y,z ) ⇒ P ( x,z )] A 3 : ∀ x ¬ P ( x,x ) Assume that M = h M,P M i | = A 1 ∧ A 2 ∧ A 3 . What can you say about the cardinality of M ? 1 (10%) 6. Show that the formula ( ∃ xA ∧ ∃ xB ) ⇒ ∃ x ( A ∧ B ) (*) is in general not logically valid (i.e., find a language L and formulas A,B in L such that the formula ( * ) is not logically valid). (10%) 7. Show that if S is a binary relation symbol, then | = ¬∃ y ∀ x ( S ( y,x ) ⇔ ¬ S ( x,x )) . (10%) 8. Let M = h N ,< i , where < is the usual ordering on N , and let S = h N ,< S i , where < S is the following binary relation on N : n < S m iff ( n,m are both even or both odd, and n < m ) or ( n is even and m is odd). Find a sentence A in the language L = { < } , where < is a binary relation symbol, such that M | = A but S | = ¬ A. 2...
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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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09ma6cHW6 - A 1 ∀ x ∃ yP x,y A 2 ∀ x ∀ y ∀ z P...

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