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1. Symmetric difference; 2. Commutativity of natural numbers

  1. Aug 6, 2012 #1
    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!
     
    Last edited: Aug 6, 2012
  2. jcsd
  3. Aug 6, 2012 #2

    HallsofIvy

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    For (1) you don't prove "the existence of a unique set [itex]A'[/itex]". 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.
     
    Last edited: Aug 7, 2012
  4. Aug 6, 2012 #3
    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 [itex]A'[/itex] such that symmetric difference is not empty right?
     
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