for (1): note that means and . clearly we cannot have if . so all you have to do, is show that if (as i suggested you start with) then ... oh, just let me do it.
for the first implication, assume . Then . Clearly if , then . thus, we have that and . It means therefore, that is not in their intersection. so, .
for the second implication, you need to show that .
now, if , then there exists an such that .
for problem (3):
Let be defined as i said.
by induction, we need to prove is true, then show that if is true, then will be true.
Clearly is true. since when , . and, of course, is divisible by 8.
Assume is true for some . we show is true.
Since is true, we can write for some integer .
so, we need to show we can write in this fashion.
for , we have:
now show that we can write this as for some integer , given that
on second thought, i don't like my proof for the first implication in problem (1). there seems to be a problem with it.
instead, do it by the contrapositive. assume and show that implies .
in other words, i want you to prove the equivalent statement that:
you need to be clear on what it means for and