I have two logic statements as follows:
$\forall a \exists b \forall c \; Sport(a,b,c)$ (for all $a$ and $c$, there exists a sport $b$ which they share)
and
$\forall a \forall c \exists b \;Sport(a,b,c)$
Now I want to know whether the two statements are equivalent, weaker or stronger than each other. When I try and "convert" these to plain english, they seem to be conveying the same thing but I don't know how to prove it.
Think about what the order of the quantifiers means with regard to the dependence of the subsequent variable on the earlier ones. Prop 1 reads " for every a there is a b such that for every c P(a,b,c)." Where Prop 2 reads "for every a and for every c there is a b." In the second statement b depends on a and b whereas in the first statement b only depends on a and then works for any c you choose.