Generating Algebras of Sets
We might not always want to lay out an entire algebra of sets in one go. Sometimes we can get away with a smaller collection that tells us everything we need to know.
Suppose that is a subset of — a collection of subsets of — and define to be the collection of finite disjoint unions of subsets in . If we impose the following three conditions on :
- The empty set and the whole space are both in .
- If and are in , then so is their intersection .
- If and are in , then their difference is in
then is an algebra of sets.
If , then , and so contains and . We can also find , since .
Let’s take and to be two sets in , written as finite disjoint unions of sets in . Then their intersection is
Each of the is in , as an intersection of two sets in , and no two of them can intersect. Thus finite intersections of sets in are again in .
If , then . Since each of the are in , their (finite) intersection must be as well, and is closed under complements.
And so we can find that if and are in , then and are both in , and is thus an algebra of sets.