1. Is \(P\) true or false? Explain your answer.
  2. Rewrite the negation \(\neg P\) such that the negation symbol does not occur in front of quantifiers or logical connectives.
  3. Is \(Q\) true or false? Explain your answer.
  4. Rewrite the negation \(\neg Q\) such that the negation symbol does not occur in front of quantifiers or logical connectives.

Answer:

  1. P is true. Every number \(x\) has some number \(y\) that is bigger, in the set \(\mathbb{N}\).
  2. Rewritten negation \(\neg P = \exists x \in \mathbb{N}, \forall y \in \mathbb{N}, x \geq y\)
  3. 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. 
  4. Rewritten negation \(\neg Q = \exists x,y \in \mathbb{N}, (x^2 < 0) \vee (y^2 < 0) \wedge x^2 + y^2 \geq 0 \)