suppose you had a very small set of numbers that contained only 0 and 1 . Would this set be closed under addition? if not give a counterexample

Answer 1

I assume your set is the set of binary numbers.
Since you supposed you had a very small set of numbers, I would conclude that there is a maximum number is your set.
suppose it is 1111
then 1001 + 11001 would not be in that set.