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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.
 Spring '09
 InessaEpstein
 Math, Logic

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