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# X y lxy lyx a how long a list of distinct structures

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x )( y )( Lxy ⊃ ¬ Lyx ) (a) How long a list of distinct structures A with universe of discourse { 1 , 2 , 3 } satisfy the schema S ? 27 (b) How long a list of pairwise non-isomorphic structures with universe of discourse { 1 , 2 , 3 } satisfy the schema S ? 7 (c) How long a list of distinct structures A with universe of discourse { 1 , 2 , 3 , 4 } satisfy the condition: A | = S and | Aut ( A ) | = 1? 18 (d) How long a list of distinct structures A with universe of discourse { 1 , 2 , 3 } satisfy the condition: A | = S and | Def ( A ) | = 2? 3

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3. (30 points) For each of the following pairs consisting of a set of schemata X and a schema S determine whether X implies S. If so, provide a deduction to establish the implication. If not, specify a structure which makes S false and all the schemata in X true. (a) X : { ( x )(( y ) Lxy ( z ) Lzx ) , ( x )( y ) Lxy } S : ( u )( z ) Lzu A : U A = L A = Deduction { 1 } (1) ( x )( y ) Lxy P { 1 , 2 } (2) ( y ) Lwy (1) w EII { 3 } (3) ( x )(( y
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x y Lxy Lyx a How long a list of distinct structures A with...

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