Difference between revisions of "Ring of sets"

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Revision as of 15:12, 13 March 2015

A Ring of sets is also known as a Boolean ring

Note that every Algebra of sets is also a ring, and that an Algebra of sets is sometimes called a Boolean algebra

Definition

A Ring of sets is a non-empty class [ilmath]R[/ilmath][1] of sets such that:

  • [math]\forall A\in R\forall B\in R(A\cup B\in R)[/math]
  • [math]\forall A\in R\forall B\in R(E-F\in R)[/math]

First theorems

The empty set belongs to every ring


Take any [math]A\in R[/math] then [math]A-A\in R[/math] but [math]A-A=\emptyset[/math] so [math]\emptyset\in R[/math]



References

  1. Page 19 - Halmos - Measure Theory - Springer - Graduate Texts in Mathematics (18)