If a signature σ is infinite (i.e. contains infinitely many symbols), then a σ-formula can have infinite length.  b. Let σ := (c, f, ·, <) be a signature, where c is a contant symbol, f is a unary operation symbol, · is a binary operation symbol, and < is a binary relation symbol. Then ϕ(y, z) := ∀x(f(x) · c < x) is an extended σ-formula whose interpretation in each σ-structure is a binary relation.  c. For any signature σ and any σ-structure A := (A, σA), if a set D ⊆ Ak is definable (A-definable here), then it is P-definable for some finite P ⊆ A.

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
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Is each of the following statements true or false? Please explain. 

a. If a signature σ is infinite (i.e. contains infinitely many symbols), then a σ-formula can have infinite length. 

b. Let σ := (c, f, ·, <) be a signature, where c is a contant symbol, f is a unary operation symbol, · is a binary operation symbol, and < is a binary relation symbol. Then ϕ(y, z) := ∀x(f(x) · c < x) is an extended σ-formula whose interpretation in each σ-structure is a binary relation. 

c. For any signature σ and any σ-structure A := (A, σA), if a set D ⊆ Ak is definable (A-definable here), then it is P-definable for some finite P ⊆ A. 

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