Negation Math Example 1

Solution

1. 1
   Recall that the negation $\neg P$ of a statement $P$ is the statement that is true exactly when $P$ is false.
2. 2
   (a) $P$: '$5 > 3$' (True). The negation reverses the inequality: $\neg P$: '$5 \le 3$' (False). (b) $P$: 'All cats are black' has form $\forall x, P(x)$. Its negation is $\exists x, \neg P(x)$: 'There exists a cat that is not black.'
3. 3
   Truth values: (a) $\neg P$ is False because $5 > 3$ is true. (b) $\neg P$ is True because black cats are not the only kind — there exist non-black cats in the world.