Find the negation of the proposition P: P: ∀xN; (x is even −→ x^2 + 1 is odd )

Answer 1

In general, suppose you have a proposition P: ∀ x; (Q(x) → R(x)), where Q,R are predicates about x. Then the negation of P is

¬ P : ∃ x. (Q(x) AND ¬R(x)).

In your example, we can define Q(x) := "x is even", and let R(x) := "x² + 1 is odd". So the negation of P is

¬ P : ∃ x; ((x is even) AND (x² + 1 is even) ).

I hope this helps!