# 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 $$G$$?

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

The collection G of sets is closed under intersections and complements.

• Is that entirely correct, though? For example, it doesn't contain $E_1 \cap E_1^{'}$ Also, suppose my k was larger, such as 3. Then $G$ would contain $E_1 \cap E_2 \cap E_3$ but not $E_1 \cap E_2$. Commented Nov 17, 2017 at 5:30
• @JohnTravolski It does contain $E_1 \cap E_1'$ - see my answer. Either that or you have to rephrase more carefully what your system is supposed to be. Commented Nov 17, 2017 at 5:35

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.) .