Construction of the Null Set

Making nothing out of something

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Let X be a set with at least two members, and S be the collection of its nonempty subsets. Let Y=Stimes S, the collection of all ordered pairs of nonempty subsets of X. Define an equivalence relation on Y satisfying left( A,B right)sim left( C,D right)  if and only if 

  1. A and B have no common members and C and D have no common members, or
  2. Acap B=Ccap D.
Define Z to be the collection of equivalence classes induced by sim. Denote by overline{left( A,B right)} the equivalence class containing left( A,B right). There is a natural injection between S and Z given by the mapping Amapsto overline{left( A,A right)}, so that we may identify each nonempty subset Asubseteq X with the equivalence class overline{left( A,A right)}in Z. This leaves one unmapped equivalence class in Zoverline{left( A,A^c right)} where A is any nonempty proper subset of X and A^c is the complement of A, i.e. Xbackslash A. This equivalence class we identify as the null set, and denote it by emptyset,
Set Operations Under the Identifications
  1. overline{left( A,B right)} cap emptyset=emptyset cap overline{left( A,B right)}=emptyset.
  2. For overline{left( A,B right)} ne emptyset and overline{left( C,D right)} ne emptyset, we define overline{left( A,B right)} cap overline{left( C,D right)}=overline{left( A cap B,C cap D right)}.
  3. overline{left( A,B right)} cup emptyset=emptyset cup overline{left( A,B right)}=overline{left( A,B right)}.
  4. For overline{left( A,B right)} ne emptyset and overline{left( C,D right)} ne emptyset, we define overline{left( A,B right)} cup overline{left( C,D right)}=overline{left(left( A cap B right) cup left(C cap D right),left( A cap B right) cup left(C cap D right) right)}.
  5. overline{left( X,Xright)}^c=emptyset
  6. overline{left( A,Xright)}^c=overline{left( X,Aright)}^c=overline{left( A,Aright)}^c=overline{left( A^c,A^cright)}.
  7. For A ne X and B ne X, we define overline{left( A,B right)}^c=overline{left( A^c cup B^c,A^c cup B^c right)}.
An Illustrative Example

Take X=left{a,b right} . Then Y={left(left{aright},left{aright} right) ,left(left{aright},left{bright} right) ,left(left{aright},left{a,bright} right), left(left{bright},left{aright} right) ,left(left{bright},left{bright} right) ,left(left{bright},left{a,bright} right), left(left{a,bright},left{aright} right) ,left(left{a,bright},left{bright} right) ,left(left{a,bright},left{a,bright} right)}. The equivalence classes are:
left{ left( left{ a right},left{ a right}  right) , left( left{ a right},left{ a,b right}  right) , left( left{ a,b right},left{ a right}  right) right}  which corresponds to left{ a right};
left{ left( left{ b right},left{ b right}  right) , left( left{ b right},left{ a,b right}  right) , left( left{ a,b right},left{ b right}  right) right}  which corresponds to left{ b right};
left{ left( left{ a,b right},left{ a,b right}  right)  right}  which corresponds to left{ a,b right}; and 
left{ left( left{ a right},left{ b right}  right) , left( left{ b right},left{ aright}  right)right}  which corresponds to the null set.