Question

is the set of (-1,0,1) a closed set under addition, subtraction, and division? why or why not?

Answer 1

Yes, it is to addition and multiplication, because if you sum any of the numbers of the set (like -1+0=-1, -1+1=0, 0+1=1) you will always get a number that is already in the set.
Same with multiplication, ex: -1x0=0 (already in the set), -1x1=-1(already in the set), etc
But it isn't closed to division or substraction because if you divide -1/0 or 1/0 you dont get a number of the set (indeed is undefined), or 1-(-1)=2 and 2 isnt in the set.

Answer 2

What if you add \(1\) to \(1\)? Any operation you choose isn't necessarily restricted to only being applied to distinct elements.