Consider a symbol vocabulary t

Consider a symbol vocabulary that contains c constant symbols, predicate symbols of each arity k, and fk function symbols of each arity k, where 1 ≤ k ≤ A. Let the domain size be fixed at D. For any given interpretation—model combination, each predicate or function symbol is mapped onto a relation or function, respectively, of the same arity. You may assume that the functions in the model allow some input tuples to have no value for the function (i.e., the value is the invisible object). Derive a formula for the number of possible interpretation— model combinations for a domain with D elements. Don’t worry about eliminating redundant combinations.

 

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