Predicate Calculus help

Question

Working on predicate calculus this week, and was hoping I've got these correct, but I'm sure I've made some mistakes for sure..

1. All programmers enjoy discrete structures
2. not all integers are odd
3. every integer that is divisible by two is even

4. there exists a natural number that is not positive.

   ---

   1) $x =$ programmer

   $f(x) = x$ enjoys discrete structures

   $∀x f(x)$

   ---

   2) $x =$ integers

   $O(x) = x$ is odd

   $\sim∀x O(x)$

   ---

   3) $x =$ integer

   $b(x) = x$ is divisible by two

   $c(x) = x$ is even

   $∀x b(x) ⇒ c(x)$

   ---

   4) $x =$ natural number

   $f(x)=x$ that is positive

   $∃x \sim f(x)$

Answer 1

You seem to be on the right track. I'll give some constructive criticism:

(a) The first line shouldn't be declaring what $x$ is. You should be defining a set. For example, for the first question, your first statement should be something like "Let $S$ be the collection of all programmers". This ties immediately into

(b) When you use a quantifier, immediately following that quantifier you should declare an element belonging to a set. Don't say $\exists x$, say $\exists x\in W$, where $W$ is some set you have defined.