1. The problem statement, all variables and given/known data I have two problems that I got stuck. 1. [itex]\exists ! N\in P(X) , A\Lambda N=A, \forall A\in P(X)[/itex] and for each [itex]A\in P(X), \exists ! A'\in P(X) , A\Lambda A' =N [/itex] 2. Prove [itex]a+(b+c) = (a+b) +c[/itex], for positive integers [itex]a, b, c[/itex] 2. Relevant equations 1. Given sets [itex]A,B \in P(X)[/itex], where [itex]P(X)[/itex] denotes power set, their symmetric difference is defined by [itex] A\Lambda B= (A - B)\cup (B-A) = (A\cup B) - (A\cap B)[/itex] 2. For the second exercise, I am asked to use this definition: The sum [itex]m+n[/itex] of positive integers [itex]m, n[/itex] may be defined by induction on [itex]n[/itex] by [itex] (i) m+1=s(m) [/itex] [itex] (ii)\forall k\in Z^+, m+s(k)=s(m+k) [/itex] where [itex]s(m)[/itex] is successor function 3. The attempt at a solution For the first part of first exercise, I think I proved that the only such set [itex]N[/itex] is the empty set. But then how should I proceed to proving the existence of unique set [itex]A'[/itex] ? For the second exercise I do not know how exactly I should begin. Should I use induction for one of the numbers? For all of them? The thing is I am not yet used to putting everything in mathematical language which gets difficult proving simple fundamental properties, so I need your hints guys. Thanks!