Let X be a set, and define a (Boolean) sigma-ring _S_ as a non-empty
class of subsets of X that is closed under the formation of differences
and countable unions.
Let _E_ be any class of subsets of X, and denote by _S_(_E_) the
smallest sigma-ring containing _E_.
Let A be a subset of X, and denote by E the generic element of the class
_E_. Denote by intersection(_E_,A) the class of sets
{intersection(E,A): E in _E_}. Denote by _S_(intersection(_E_,A)) the
smallest sigma-ring containing the class of sets intersection(_E_,A).
Based on these definitions, create a class of sets _C_ to be {union(B,
diff(E,A)): B in _S_(intersection(_E_,A)), E in _S_(_E_)}. In other
words, each element of the class _C_ is the union of an element B of
_S_(intersection(_E_,A)) with the difference of an element E of _S_(_E_)
and the set A.
How do you prove that the class _C_ is a sigma-ring? (this is supposed
to be "easy") I managed to prove that the simpler class {diff(E,A):E in
_S_(_E_)} is a sigma-ring, but can't find any way to prove that _C_
itself is a sigma-ring.
Thanks,
-sto
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