Translate the following inference to logic notation: For all x, z_4 is a janitor or z_4 will not write a letter. There is at least one x for which z_4 is a janitor if and only if z_4 will write a letter. Therefore there is at least one x for which z_4 will write a letter if and only if z_4 is a janitor.
forall x: P(x) or ~Q(x). exists x: P(x) <-> Q(x). Therefore exists x: Q(x) <-> P(x).