Resources: Chapter 8 of *HLW*

For each of the following sequents, provide a counterexample to show that it is invalid.

∀

*x**F**x*→ ∀*x**G**x*⊢ ∀*x*(*F**x*→*G**x*)∀

*x*(*F**x*→*G**x*) ⊢ ∃*x*(*F**x*∧*G**x*)⊢ ∀

*x**F**x*∨ ∀*x*¬*F**x*∃

*x*(*F**x*→*P*) ⊢ ∃*x**F**x*→*P*

For each of the following sentences, provide one interpretation in
which it is true and one interpretation in which it is false. An
interpretation may be presented by giving a set *M* and a subset *R*^{M} of *M* × *M*, or it may be
presented as an arrow diagram.

∀

*x*∀*y*∃*z*(*R**x**z*∧*R**y**z*)∀

*x*(∃*y**R**y**x*→∀*z**R**z**x*)