Section 2.1: Problem 8 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
In 3–8, translate each English sentence into the first-order language specified. (You may want to carry out the translation in several steps, as in some of the examples.) Make full use of the notational conventions and abbreviations to make the end result as readable as possible.
(a) Every farmer who owns a donkey needs hay. (b) Every farmer who owns a donkey beats it. ($\forall$ , for all things; $F$ , is a farmer; $D$ , is a donkey; $Oxy$ , $x$ owns $y$ ; $H$ , needs hay; $Bxy$ , $x$ beats $y$ .)
(a) $\forall x(Fx\wedge\exists y(Dy\wedge Oxy)\rightarrow Hx)$ . Alternatively, $\forall x(Fx\rightarrow\exists y(Dy\wedge Oxy)\rightarrow Hx)$ .
(b) $\forall x(Fx\rightarrow\forall y(Dy\wedge Oxy\rightarrow Bxy))$ .