Question

let p, q be natural numbers and P(p,q) be some statement true for some pairs p,q but not for others

does this being true:

∃p∀q:P(p,q)


imply that the following statement is true:

∀q∃p:P(p,q)

Answer 1

can you be my @Hero and help me?

Answer 2

@amistre64

Answer 3

@AccessDenied

Answer 4

im sorry for the tagging spree

Answer 5

@bahrom7893

Answer 6

Let me make sure I understand the statement:
Does "There Exists some p for All q such that P(p,q) is true" imlpy that
"For All q, There Exists some p such that P(p,q) is true"

Answer 7

surely it does

Answer 8

Because it sure seems like it to me.

Answer 9

on my worksheet it asks for a counter example of this statement and i haven't found one. also i believe a proof is available

Answer 10

If I translated it correctly, I can't see a counterexample.

Answer 11

Nevermind you're right.

Answer 12

There Exists some p for All q such that P(p,q) is true

let this value of p be denoted as p*

now

For All q, There Exists some p* such that P(p,q) is true

Answer 13

im confused though, my worksheet asks effectively for a counterexample

Answer 14

The latter does not imply the former, however, so make sure you read it properly.

Answer 15

Is the empty set a counter example?

Answer 16

Or is the counterexample the empty set?

Answer 17

here is the original question (its part (b) )

Let P(p; q) denote a statement about natural numbers p and q which is true for
some pairs p and q and false for other pairs. For example, P(p; q) might be the statement
`p < q^2 ' or the statement `q = 10^(23p) + 42'. Give examples in which
(a) ∀p∃q P(p; q) is true but ∃q∀p P(p; q) is false;
(b) ∃p∀q P(p; q) is true but ∀q∃p P(p; q) is false.

\[\exists \forall\]

Answer 18

ignore the two symbols at the bottom

Answer 19

The first is incorrect... for all p there exists a q such that P(p;q) does not imply that there exists q such that for all p P(p;q). The second looks fine to me though.

Answer 20

yeah an example for the first is pq = q^2