i was only talking about your first sentence of number two I assumed that when you wrote {} you where speaking about the empty set and nothing else you said the proof is imediate
i said elsewhere recently that you can proof anything you want about the element of the empty set and the contrary
i agree when you say if a and b belongs to S (the empty set) it implyes that a*b belongs to S this is a true statement because they are no a and b belonging to S
but you can proof the same way that s (the empty set) is unclosed a*b would not belong to S even if a and b where both belonging to S