- Is \(P\) true or false? Explain your answer.
- Rewrite the negation \(\neg P\) such that the negation symbol does not occur in front of quantifiers or logical connectives.
- Is \(Q\) true or false? Explain your answer.
- Rewrite the negation \(\neg Q\) such that the negation symbol does not occur in front of quantifiers or logical connectives.
Answer:
- P is true. Every number \(x\) has some number \(y\) that is bigger, in the set \(\mathbb{N}\).
- Rewritten negation \(\neg P = \exists x \in \mathbb{N}, \forall y \in \mathbb{N}, x \geq y\)
- Q is true. The expression \((x^2 < 0) \vee (y^2 < 0)\) will always be false. The expression \(x^2 + y^2 < 0\) will also always be false, which makes the implication always false. If the boolean value before the implication is false, the whole expression will be true.
- Rewritten negation \(\neg Q = \exists x,y \in \mathbb{N}, (x^2 < 0) \vee (y^2 < 0) \wedge x^2 + y^2 \geq 0 \)