Write propositions strictly using quantifiers

Question

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?

Answer 1

As an example for the first question

Define predicates such that

$C(x):x$ is a student of Calculus.

$A(x):x$ is an athlete.

All the students of Calculus are athletes.

$$\forall x(C(x)\to A(x))$$ Try fill the blanks

$F(x):x$ is a fish

$G(x):x$ has gills

Each fish has gills.

$$\forall x(?~\to~?)$$

$D(x):x$ is a dog.

$S(x):x$ has spots.

Some dogs have spots.

$$\exists x(?~\land~?)$$

Answer 2

Let us speak the set Theory language.

We denote the sets of students of calculus and athletes by S and A.

then, we will say

$$(\forall x\in S) \;\; x\in A$$

or if we call E, the set of all students, we could write

$$(\forall x\in E)\;\; (x\in S \implies x\in A)$$