Translate the following inference to logic notation: There is at least one x for which x_8 will take a plane if and only if x_8 will sleep. For all x, if x_8 takes a plane, then x_8 is writing a letter. For all x, if x_8 writes a letter, then x_8 is taking a plane. For all x, x_8 will not sleep. Therefore there is at least one x for which x_8 will not write a letter.
exists x: P(x) <-> Q(x). forall x: P(x) -> R(x). forall x: R(x) -> P(x). forall x: ~Q(x). Therefore exists x: ~R(x).