(a) A o-formula is called universal (resp. existential) if it is of the form \x₁\x...\x (resp. 3x₁x₂...3x), where & is quantifier free, i.e. does not contain quantifiers 3, V. Let A, B be ♂-structures with AC B and let (7) be a σ-formula. Show that for any đe A”, (i) if p is quantifier free, then A = y(a) ⇒ B‡q(ā); (ii) if is universal, then B = y(@) ⇒ A = y(a); (iii) if p is existential, then A = o(a) ⇒ B= y(a). (b) Find a sentence that is true in (IN, <) but false in (Z, <).
In mathematical logic, structures provide a context where formulas can be interpreted and checked for their truth values.
Given σ-formulas and σ-structures, we are working within a specific language or signature, σ.
A formula is said to be quantifier-free if it does not contain any universal (∀) or existential (∃) quantifiers.
When we talk about one structure being a subset of another, A ⊆ B, it means every element in A is also in B, and every relation or function in A is also in B, but B might have more.
The symbols "⊨" denote the satisfaction relation, which says that a particular structure satisfies a given formula.
The crux of the problem is to understand how the truth values of certain formulas might change (or remain the same) when interpreted over larger structures.
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