I'm struggling to figure out the difference between these two statements:
∀a ∈ A : ∃b ∈ B so that something holds, and ∃b ∈ B : ∀a ∈ A so that something holds.
We're asked to give an example but I'm looking to clarify the distinction. Thanks all!
2026-03-28 09:44:18.1774691058
Order in a First Order Logic Statement with Quantifiers?
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$\forall a\in A:\exists b\in B$ is also expressed as $\forall a~\exists b~(a\in A\to b\in B)$.
This is satisfied when something is in $B$ or nothing is in $A$.
$\exists b\in B:\forall a\in A$ is also expressed as $\exists b~\forall a~(b\in B\wedge a\in A)$.
This is satisfied when something is in $B$ and everything is in $A$.