We are asked to determine if the infinite set {0,1,2,3,...} is closed under addition and to find the inverse.

A set is closed under an operation if when we take two elements of the set and apply the operation to them the result is also a member of the set. If you add any nonnegative integer (natural number) to another natural number (some texts do not allow 0 as a natural number—if this is the case use nonnegative integer) we get another natural number. Thus this set is closed under addition. (The set is not closed under subtraction as 2-3 is not a member of the set.)

Under addition of integers, 0 is the identity. (a+0=0+a=a for a a nonnegative integer.) The inverse of addition is subtraction since a - a = 0 where 0 is the identity for addition.