Is this set closed under addition or multiplication or both and why? $\{-1,0,1\}$
Please give an explanation and also tell me what does closed under addition and multiplication mean. 
Different definitions are given everywhere.

Let's go through 2 examples here and I'll show you how that set could be considered closed or not closed depending on the definition of the operations:
Integer mod 3 example:
In this case, there is {1,0,-1} which is equivalent to {1,0,2} mod 3 as 2=-1 in this case.  In this case, the set is closed as all the operations stay in the field.  In this case, 1+1=2 which is still in the field. 2+2 =4= 1 as remember that the equivalence here is mod 3, so a=b if there exists an integer k such that a = 3k+b.
Natural numbers case:
In this case, it isn't closed as 1+1=2 which isn't in the set.
This is why defining the operations is rather important as while most people can infer what addition and multiplication mean, they can be redefined in some cases.
