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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., nd 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 i ( 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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 Spring '09
 InessaEpstein
 Math, Logic

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