If a statement says that a set is empty, then its negation is the statement that the set is nonempty. That is straightforward. Suppose a statement says “something-happens” for all $i\text{,}$ or every $i\text{,}$ or any $i\text{.}$ Then the negation is that “something-does-not-happen” for at least one value of $i\text{.}$ If a statement says that there exists at least one “thing,” then the negation is the statement that there is no “thing.” If a statement says that a “thing” is unique, then the negation is that there is zero, or more than one, of the “thing.”