If I want to translate the following sentences to predicate logic:
All students enjoy math.
is it ok to specify $x$ are students and just do the following?
$$M(x) = x \text{ enjoys math}$$
so
$$\forall x (M(x))$$
Negation:
∃x (~M(x)) = There exists a student that does not enjoy math.
Or do I have to use another predicate to specify what x is?
S(x) : x is a student. and then, M(x) : x enjoys math.
∀x(S(x)→M(x)) , which should negate to ∃x(S(x) ∧ ~M(x)) . Similar here, but when statements get more complicated, the negations turn out to be slightly different for the two methods.
Are both acceptable?