Quantifiers
A rule holds for every number, or for at least one. How do you say which, exactly, and what does each take to prove?
Where you are 4 inside
What this area is about
A sentence like x^2 = 2 is not true or false until you know what x is. A quantifier settles that by saying how many members of a domain make it true: all of them, at least one, or exactly one. The same sentence can flip from true to false when only the domain changes.
How these fit together
The two quantifiers. For all, \forall, and there exists, and there is exactly one, \exists and \exists!, are the definitions everything else is built from. “For all” and “there exists” are lopsided mirror images: one counterexample breaks a “for all”, and one witness proves a “there exists”. “Exactly one” is “there exists” plus the claim that no second witness exists, so it shares that page.
The procedure. Negating a quantified statement turns each \forall into \exists and each \exists into \forall, and moves the “not” onto the sentence inside.
Combining them. Nested quantifiers put one inside another, and there the order matters: \exists y\ \forall x always gives \forall x\ \exists y, but not the other way round.
A path through it
For all, then there exists and exactly one, then negation, then nesting.
Finish here and you can read any claim shaped “every ... has some ...” and say what would prove it and what would break it. That is what a counterexample needs, and it is where existence proofs and uniqueness proofs start.