Translate the following inference to logic notation: For all x, x_8 will listen to a song if and only if x_8 will write a letter. There is at least one x for which x_8 will write a letter or x_8 will sleep. For all x, x_8 will listen to a song if and only if x_8 will sleep. Therefore there is at least one x for which x_8 will listen to a song.
forall x: P(x) <-> Q(x). exists x: Q(x) or R(x). forall x: P(x) <-> R(x). Therefore exists x: P(x).