(a) A o-formula is called universal (resp. existential) if it is of the form \x₁\x₂...\x₁(resp. 3x₁3x2...³xµ½), where & is quantifier free, i.e. does not contain quantifiers 3,V. Let A, B be o-structures with A CB and let (7) be a o-formula. Show that forany de A¹,(i) if p is quantifier free, then A = y(a) ⇒ B=¶(ā);(ii) if p is universal, then B = o(a) ⇒ A = y(a);(iii) if is existential, then A = o(a) ⇒ B= y(a).(b) Find a sentence that is true in (N, <) but false in (Z,<).

In mathematical logic, structures provide a context where formulas can be interpreted and checked…

(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, <).

(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, <).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.5: Graphs Of Functions

Problem 56E

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