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
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.
To determine if a set is closed under addition, we need to check if the sum of any two elements in the set also belongs to the set.
In this case, the set contains only 0 and 1. Let's consider all possible additions of 0 and 1:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 2
Since the result of 2 is not in the given set, we can conclude that the set is not closed under addition.
Counterexample: The numbers 0 and 1 form a set that is not closed under addition.