Write the following propositions strictly using quantifiers and give also strictly the negation of the propositions.
- All the students of Calculus are athletes.
- Each fish has gills.
- Some dogs have spots.
Firstly, taking the first proposition into consideration, I have thought that it is equivalent to saying that if $x$ is a student of Calculus, then $x$ is an athlete.
But how do we use the quantifiers?
I have thought of the following:
$\forall x$:$x$ is a student of Calculus, $x$ is an athlete.
But is this enough? Or do we have to build a proposition with more quantifiers ?
Could you give a hint so that I tell you also my efforts for the other two propositions?