4 let s 2 be the conjunction of the following

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4. Let S 2 be the conjunction of the following schemata. ( x )( y )( Lxy Lyx ) ( x )( y )( z )(( Lxy Lyz ) Lxz ) ( x ) Lxx (a) (10 points) Specify a structure A 2 of size at least 4 which satisfies S 2 , that is, U A 2 has at least 4 members and A 2 | = S 2 . U A 2 =
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L A 2 = (b) (10 points) How many structures with universe of discourse { 1 , 2 , 3 , 4 } satisfy S 2 ? 5. Let S 3 be the conjunction of the following schemata. ( x )( y )( Lxy ⊃ ¬ Lyx ) ( x )( y )( z )(( Lxy Lyz ) Lxz ) ( x )( y )( x 6 = y ( Lxy Lyx )) ( x )( y )( Lxy ( w )( Lxw ( w = y Lyw )) ( x )(( y ) Lyx ( y )( Lyx ( z )( Lyz Lzx ))) (a) (10 points) Specify a structure A 3 of size at least 4 which satisfies S 3 , that is, U A 3 has at least 4 members and A 3 | = S 3 . U A 3 = L A 3 = (b) (10 points) How many structures with universe of discourse { 1 , 2 , 3 , 4 } satisfy S 3 ? 6. We say that a schema S admits a positive integer n if and only if there is a structure A of size n which satisfies S ; the spectrum of S is the set of positive integers n such that S admits n . (a) (10 points) What is the spectrum of the conjunction of the following schemata? ( x )( y )( Lxy ⊃ ¬ Lyx ) ( x )( y )( z )(( Lxy Lyz ) Lxz ) ( x )( y )( x 6 = y ( Lxy Lyx )) ( x )(( y ) ¬ Lyx ( y ) ¬ Lxy ) Fx ) ( x )( y )(( Lxy ( z ) ¬ ( Lxz Lzy )) ( Fx ≡ ¬ Fy )) (b) (10 points) What is the spectrum of the conjunction of the following schemata?
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( x ) ¬ Lxx ( x )( y )( Lxy Lyx ) ( x )( y )( z )( Lxz y = z )
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