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?