1. Symmetric difference; 2. Commutativity of natural numbers

[threeder]

Homework Statement

I have two problems that I got stuck.
1. $\exists ! N\in P(X) , A\Lambda N=A, \forall A\in P(X)$ and for each $A\in P(X), \exists ! A'\in P(X) , A\Lambda A' =N$

2. Prove $a+(b+c) = (a+b) +c$, for positive integers $a, b, c$

Homework Equations

1. Given sets $A,B \in P(X)$, where $P(X)$ denotes power set, their symmetric difference is defined by $A\Lambda B= (A - B)\cup (B-A) = (A\cup B) - (A\cap B)$

2. For the second exercise, I am asked to use this definition:
The sum $m+n$ of positive integers $m, n$ may be defined by induction on $n$ by
$(i) m+1=s(m)$
$(ii)\forall k\in Z^+, m+s(k)=s(m+k)$
where $s(m)$ is successor function

The Attempt at a Solution

For the first part of first exercise, I think I proved that the only such set $N$ is the empty set. But then how should I proceed to proving the existence of unique set $A'$ ?

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!

 

[HallsofIvy]

For (1) you don't prove "the existence of a unique set $A'$". The problem is to prove the statement is NOT true for any set N.

For (2), use induction on c to prove that, for any fixed a and b, there exist c such that (a+ b)+ c= a+(b+ c). By the way, this is associativity, not commutativity. I assume that was a typo.

 

[threeder]

Indeed it was a typo :) Anyhow, I managed to deal with the second.

Concerning the first, could you verify that N=∅ for the first part of the first exercise? In that case all I need is just a counterexample to show that there exists one $A'$ such that symmetric difference is not empty right?