Name for set of sets $\bigcap_{i=1}^k A_i$ where $A_i$ is either $E_i$ or its complement Suppose there are events $E_i$ for integer values such that $1\leq i\leq k$.  Now let some set of sets (call it $G$) be composed of every set that is the intersection of all $k$ events or (exclusive or) their complements.  I'm finding it difficult to word this correctly, so this should hopefully help:
For example, if $k=2$, then $G$ should ONLY contain:
${E_1 \cap E_2}$,
${E_1 \cap E_2^{'}}$,
${E_1^{'} \cap E_2}$,
${E_1^{'} \cap E_2^{'}}$
Where the apostrophe denotes the event's complement.  Is there a name for such a set $G$?
 A: We could call it the partition generated by the class $\{E_i\}_{i=1}^k$.
Some examples of this usage:


*

*Dorian Feldman and Martin Fox, Probability: The Mathematics of Uncertainty, Section 4.B: Partition Generated by a Class

*Paul Pfeiffer, Concepts of Probability Theory, Definition 2-7b

*Achim Klenke, Probability Theory, proof of Hahn's decomposition theorem:


Define $A_n^0 := A_n$, $A_n^1 := A \setminus A_n$, and let $$\mathcal P_n := \left\{ \bigcap_{i=1}^n A_i^{s(i)} : s \in \{0,1\}^n\right\}$$ be the partition of $A$ that is generated by $A_1, \dotsc, A_n$.


*Math.SE: Partition generated by a class of subsets
A: The collection G of sets is closed under intersections and complements.
A: Observe that your system is closed under complementation since for a given $E$ in your system, $E' = E' \cap E'$ is also in your system. 
Since your system is closed under complementation and finite intersection, it's also closed under finite union. Why? Let $E,F$ be in your system. Then
$$
E \cup F = (E' \cap F')'
$$
is in your system as well.
Unless your system is empty, it contains the emptyset (since it is closed under complementation and intersection) and thus is also contains $A$. This sort of system is known as an algebra and well studied in mathematics. (Compare to $\sigma$-algebra but we don't require algebras to be closed under countable unions or intersections.)
.
