I'm taking a course in mathematical logic, it has recently covered quantifiers and now I find myself somewhat confused whether or not the existential quantifier is necessary - consider this example:
Disclaimer: Assume that our universe is the set of natural numbers
$x$ is an even number
Now, I can't really tell the difference between these two notations:
$$(\exists m)(x=2m)$$ And this: $$x=2m$$ Are they basically the same? If so, does it mean that we can always omit the existential quantifier?
Now, let's consider another example:
$x$ is prime
The same problem - I have two notations and can't choose which of them is the right one.
$$(\forall a)(a |x \Rightarrow ((a=1 \lor a=x) \land x\ne1)) $$ And $$a|x \Rightarrow ((a =1 \lor a=x) \land x\ne 1) \\ $$ And finally, the last predicate:
$x$ is composite
Is it enough to write $$ \neg $$ or it is necessary to write $$(\exists a)(\neg )$$ Do these predicates actually mean the same thing?