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