I'm in Discrete Math and I copied down some rules in my notes. Unfortunately I'm not sure if I made a typo or not, let me show you what I mean.
Equivalence of Quantified Predicates Symmetry of 'All' and 'Exist':
(All x, All y, P(x, y)) = (All y, All x, P(x,y)) [Equivalent]
(Exist x, Exist y, P(x, y)) = (Exist y, Exist x, P(x, y)) [Equivalent]
(All x, Exist y, P(x, y)) =? (Exist y, All x, P(x, y)) [Equivalent Sometimes]
The last one is what confuses me, it's supposed to be equivalent under the right domain, but I'm having trouble understanding why the order of the parameters matters. Are these notes correct? Would someone mind giving me an example illustrating why the last two are not always equivalent?
Thanks!
Example of non-equivalence:
Domain is all people.
$L(x, y):\quad$ "x loves y".
$\forall x \exists y(L(x, y))\quad $ Everyone loves someone.
$\exists y \forall x (L(x, y)) \quad$ There is someone who is loved by everyone.
In the first, for each person x, there is someone y such that $L(x, y)$. That someone may vary, depending on each $x$. I love John, you love Mary, we all love at least someone.
In the second, there is some person $y$ who everyone loves. E.g., I love Bob, you love Bob, our moderators love Bob, we all love that same Bob. (There is someone (suppose that someone is Bob, e.g.) whom everyone loves.)