Negation of statements

Statement

Write the negation of each statement.

1. The apple is my favorite fruit.
2. $5 > -2$.
3. All cats are black.
4. The function $g$ is continuous at every point.

Solution

1. The apple is not my favorite fruit.
2. $5 \le -2$ (the negation of $>$ is $\le$).
3. The statement has the form $\forall x\,P(x)$; its negation is $\exists x\,\lnot P(x)$: there exists (at least) one cat that is not black.
4. Likewise: there exists (at least) one point where $g$ is not continuous.