Question

Show that the two statements \(\neg \exists x\forall yP(x,y)\,and\,\forall x\exists y\neg P(x,y)\)where both quantifiers over the first variable in P(x, y) have the same domain, and both quantifiers over the second variable in P(x, y) have the same domain, are logically equivalent.

Short answer

The two statements where both quantifierscan be shown.

Step-by-step solution

Quantifiers over first variable

Both quantifiers over the first variable in P(x,y) will have the same domain and both quantifiers over second variable in P(x,y) have same domain.

Statements are logically equivalent

The statements are logically equivalent. Now, \(\neg (\exists x\forall yP(x,y)) \to (\neg \forall yP(x,y) \to \forall x\exists y(\neg P(x,y) \leftrightarrow \forall x\exists y\neg P(x,y)))\)

\(\neg \exists x\forall yP(x,y)\) is logically equivalent to \(\forall x\exists y\neg P(x,y).\)