Consider the propositional functions P(x) = "x plays videogames" and Q(x,y) = "x and y speak over voice chat" where the domains are students in our class. Write the statement "There are two students in the class that play video games and between them both, have spoken with every other student over voice chat." using quantifiers.
$\exists x \exists y (P(x) \land P(y) \land Q(x, y))$
$\exists x \exists y (P(x) \land P(y) \land \forall z (Q(x, z) \lor Q(y, z))$
$\exists x \exists y (P(x) \land P(y) \land \forall z (Q(x, z) \land Q(y, z))$
$\exists x \exists y (P(x) \land P(y) \land \exists z (Q(x, z) \lor Q(y, z))$