Translate the following inference to logic notation: For all x, if x_9 listens to a song, then x_9 is a scientist. For all x, x_9 will listen to a song or x_9 is not a scientist. For all x, if x_9 were a scientist, then x_9 is listening to a song. Therefore for all x, x_9 is a scientist if and only if x_9 will listen to a song.
forall x: P(x) -> Q(x). forall x: P(x) or ~Q(x). forall x: Q(x) -> P(x). Therefore forall x: Q(x) <-> P(x).