Given $F(x,y)$ and ignoring whether or not it is true, determine if the following implications are true:
a) $\forall x \exists y F(x,y) \implies \exists y \forall x F(x,y)$
b) $\exists y \forall x F(x,y) \implies \forall x \exists y F(x,y)$
My answer:
Part a would be false because the second part of the implication wouldn't necessarily be true if all of x relates to at least 1 y variable from the first part. Part b I am unsure but I think this is true since you are making the relation more broad in the second part of the implication.
Does my logic make sense?
Yes you are correct. In the second example, given $x$, one can simply choose the $y$ that works for all $x$.