I have some fundamental questions about nested quantifiers:
$$\begin{array}{c|ccc}P&1&2&3\\\hline1&T&F&T\\2&T&F&T\\3&T&T&F\end{array}$$
Is $\exists x \forall y\ P(x, y)$ the same as $\exists x \forall y\ P(y, x)$?
What determines the order in which the evaluation happens is it the variables in the parenthesis or is it the order of quantifiers regardless of parenthesis order? If so then how do I interpret that in $\exists x \forall y\ P(y, x)$, $\exists x$ goes first but it is the second variable?
Does existential "player" always wants True and Universal always wants False regardless of the position/order?
For Universal player to pick a column does it have to all be T or F? Or can they pick a mixed column (which would result in a False truth value for their pick)?