Applied Logic for Computer Scientists by Mauricio Ayala-Rincón & Flávio L. C. de Moura

Author:Mauricio Ayala-Rincón & Flávio L. C. de Moura
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Exercise 34

Prove that there is no predicate formula that holds exclusively for all finite interpretations.

Exercise 35

Let E be a binary predicate symbol, e a constant and and be binary and unary function symbols, respectively. The theory of groups is given by the models of the set of formulas

Notice that according to the three first axioms the symbol E should be interpreted as an equivalence relation such as the equality. Indeed, the three other axioms are those related with group theory itself: the fourth one states the existence of an identity element, the fifth one the inverse function and the sixth one the associativity of the binary operation.

Prove the existence of infinite models by proving that for any , the structure of arithmetic modulo n is a group of cardinality n. The elements of this structure are all integers modulo n (i.e., the set ), with addition and identity element 0.

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