# Questions & Answers of First Order Logic

Question No. 2

Consider the first-order logic sentence $\style{font-family:'Times New Roman'}{F:\forall x\left(\exists yR\left(x,y\right)\right)}$. Assuming non-empty logical domains, which of the sentence below are implied by F?

$\style{font-family:'Times New Roman'}{\mathrm I.\;\exists y\left(\exists xR\left(x,y\right)\right)}$

$\style{font-family:'Times New Roman'}{\mathrm{II}.\;\exists y\left(\forall xR\left(x,y\right)\right)}$

$\style{font-family:'Times New Roman'}{\mathrm{III}.\;\forall y\left(\exists xR\left(x,y\right)\right)}$

$\style{font-family:'Times New Roman'}{\mathrm{IV}.\;\neg\exists y\left(\forall y\neg R\left(x,y\right)\right)}$