Translating Nested Quantifiers to English sentences For this entire question, please use the following propositional function:
$P(x,y)$: $x$ has sent a postcard to $y$.
Translate the following quantified propositions to English sentences. Try to use sentences
as natural as possible.
(a) $\forall x\exists y \lnot P(x, y)$
(b) $\exists x\forall y \lnot P(x, y)$
(c) $\forall x\exists y \lnot P(y, x)$
(d) $\exists x\forall y \lnot P(y, x)$
(a) Every $x$ has some $y$ to whom he has not sent a postcard.
(b) Some $x$ has not sent a postcard to every $y$.
(c) Every $x$ has some $y$ from whom he has not received a postcard.
(d) Some $x$ has not received a postcard from every $y$.
Do you think my answers are correct? 

Try the equivalent formulas in (1) and (3)
1) $\neg\exists x \forall y P(x,y)$:
There is no one that has sent a postcard to every person.
2) There is a person that has not sent a postcard to anyone.
3) $\neg\exists x \forall y P(y,x)$: There is no one that has received a postcard from every person.
4) There is a person that has not received a postcard from anyone.
