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Relations and Definability of a class of structures
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Math 461 (Spring 2020) – Homework 8
The following problems are due on Canvas by Wednesday, April 8, at 11:59pm. Please review the syllabus and Francis Su’s article “Some Guidelines for Good Mathematical Writing” (both
posted on Canvas) for expectations on how to write up your solutions. In all of these problems,
you may write formulas “informally”; just make sure that you could translate those
formulas into the “official” syntax, if necessary.
1. Let Σ be any set of L-formulas, and α and β L-formulas.
(a) Prove that Σ ∪ {α} |= β if and only if Σ |= (α → β).
(b) Prove that Σ |= ∀xα if and only if Σ |= (¬Ex(¬α)).
(c) Provide a counterexample to show that Σ |= (α ∨ β) does not imply that Σ |= α or Σ |= β.
(Hint: You need to come up with L, Σ, and α, β such that Σ |= (α∨β), and L-structures M, N
with M, N |= Σ but M 6|= α and N 6|= β.)
2. Let L be the language consisting of a single binary relation symbol R. Consider the L-sentences
ϕ : ∀v0Ev1Rv0v1, ψ : Ev1∀v0Rv0v1, and θ : Ev0∀v1Rv0v1
Which semantic implications (and non-implications) hold between ϕ, ψ and θ? For implications,
provide a proof, for non-implications, provide a counterexample.
3. Let L< be the language of ordered sets and consider the L
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