Here we want to find " A vegeterian is someone who doesn't eat meat "
$I$ says for all y if x eats y then that is not meat .
True as one is considered as vegeterian if he eats something which is not meat.
We can get $II$ from $I$
$\forall \:v(x) \Leftrightarrow (\forall y\:e(x,y) \Rightarrow \sim m(y)) \\ \forall \: v(x)\Leftrightarrow (\forall y \sim\:e(x,y)\vee\sim m(y)) \\ \forall \: v(x)\Leftrightarrow (\forall \sim\:(e(x,y) \wedge m(y))) \\ \forall x\: v(x)\Leftrightarrow \sim(\exists y \:e(x,y) \wedge m(y))$
and II says there exists a y which is meat and x eats y.
so here we got non vegetarian negating it we get vegetarian.
III is same as II as above we got non vegetarian.
For all x there exists y(meat) and x(person) eats y(meat) , then he is not vegetarian.
All are equivalent.
