Question

Let A and B be to L-structures, and let F:A→B be a bijection between their base sets. Prove that F is an isomorphism if and only if " F respects all quantifier-free formulas", that is for every quantifier-free formula φ and every assignment α with values in A, we have A⊨φ[α]⇔B⊨φ[F∘α]. Here F∘α:V→B should be thought of as the image of the assignment α under F.

Answer 1

To prove that a bijection F is an isomorphism we show that it both respects quantifier-free formulas and preserves the structure of operations and relations in L-structures A and B, in accordance with definitions of isomorphisms in the context of model theory.

Step-by-step explanation:

To prove that a bijection F between the base sets of two L-structures A and B is an isomorphism if and only if F respects all quantifier-free formulas, we consider two directions:

• If F is an isomorphism, then by definition, it preserves all operations and relations from A to B, which necessarily implies that it respects all quantifier-free formulas.
• Conversely, if F respects all quantifier-free formulas, then for every quantifier-free formula φ and assignment α with values in A, A⊨φ[α] if and only if B⊨φ[F∘α]. Since the language L of the structures is comprised of function symbols, constant symbols, and relation symbols, and quantifier-free formulas involve these without quantifiers, F must preserve these interpretations by necessity, and thus acts as an isomorphism.