Question

Is ∃x∃y(p(x) and q(y)) equivalent to [∃x(p(x) and ∃x(q(y))]?

Answer 1

I know a single existential quantifier is not distributable over conjunction but it seems they would be in this case.

Answer 2

No.

Answer 3

∃x(q(y)) doesn't exactly make sense either.

Answer 4

sorry it should be ∃x(q(x))

Answer 5

Do you have a counter example because I really can't find any. It seems to be true to me.

Answer 6

if you mean ∃x(q(x)) it might be true...

Answer 7

∃x∃y(p(x) and q(y))
=∃x∃yp(x) AND ∃x∃yq(y)
=∃xp(x) AND ∃yq(y)

Answer 8

not sure what the justification between steps 1 and 2 are, but I believe you might find something out treating ∃ as the compound OR operation

Answer 9

It seems that if ∃x∃y(p(x) and q(y)) is true. Then P(x) is true and q(y) is true. Then you can prove the converse. Seems to make sense but I'm not convinced. Anyways thanks

Answer 10

yeah if you really want to you can expand the statements out with existential generalization and existential instantiation...