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Forgive me, for I'm really lacking the proper mathematical terms to describe this, but bear with me. What I have written in words is most likely not correct; focus on my example please.

Given a set of events, {$E_1, E_2, ... E_k$}, what is the name for the set containing all combinations of intersections between exactly $k$ of those events or (exclusive or) their complements?

To be clear, this is NOT a sigma algebra. This is NOT to say that this set is "closed under intersection." This set is different. For example, if $k = 3$, then my set $G$ consists of EXACTLY:

$E_1 \cap E_2 \cap E_3$

$E_1 \cap E_2 \cap E_3^C$

$E_1 \cap E_2^C \cap E_3$

$E_1 \cap E_2^C \cap E_3^C$

$E_1^C \cap E_2 \cap E_3$

$E_1^C \cap E_2 \cap E_3^C$

$E_1^C \cap E_2^C \cap E_3$

$E_1^C \cap E_2^C \cap E_3^C$

Where the C superscript represents the complement of the event. Clearly, the set $G$ containing the 8 elements above is related to how typical truth tables are set up... but the real question is, does this set $G$ have a name? If not, how can I concisely define this set?

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You could call these the atomic sets or atoms of the sigma algebra $A=\sigma(E_1, ..., E_n)$. I think I've seen these called "elementary events" (of the sigma algebra $A$) before, but that seems to be wrong according to Wikipedia.

As you probably already know, they generate the sigma algebra $A$, which is equal to the set of unions of arbitrary sub-families of your family of sets.

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