In this problem, L is the language of equivalence relations Lear (a) Prove that the class...

 

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In this problem, L is the language of equivalence relations Lear (a) Prove that the class E of equivalence relations viewed as Lear-structures is defined by E the class of all equivalence relations such that all finite equivalence classes have even number of elements is an elementary class of Lear-structures. (b) Use Compactness Theorem I to prove that the class Efin of equivalence relations viewed as Leqr-structures defined by Efin the class of all equivalence relations which have finitely many equivalence classes is not an elementary class of Lear-structures. In this problem, L is the language of equivalence relations Lear (a) Prove that the class E of equivalence relations viewed as Lear-structures is defined by E the class of all equivalence relations such that all finite equivalence classes have even number of elements is an elementary class of Lear-structures. (b) Use Compactness Theorem I to prove that the class Efin of equivalence relations viewed as Leqr-structures defined by Efin the class of all equivalence relations which have finitely many equivalence classes is not an elementary class of Lear-structures.

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a We need to show that E is closed under isomorphism subclasses and ultralimits Closure under isomor... View the full answer