let $P(x,y): x - 7y = 5$ and $Q(x,y): x-3 \geq y + 11$
$\exists x \forall y, Q(x,y)$
$\exists x \exists y, P(x,y) \land \neg Q(x,y)$
$\forall x \exists y, P(x,y) \geq Q(x,y)$
How de we prove if these propositions are true or false, I know when a ($\forall x, P$) proposition is false you can prove it by showing that ($\exists x, \neg P$) is true, and to prove ($\forall x, P$) is true you have to use a detailed demonstration, and to prove ($\exists x, P$) you just show one example where the equation is true. However in these cases I really have no idea how to prove such propositions.