Set Theory Relations in Statistics

An event is nothing but a set of possibilities. Relationships and results from set theory can be used to study events. We use the following operations:

complement: The complement of event $A$ is denoted $A^{'}$ is the set of all outcomes in $S$ (our sample space) that are not in $A$ . Namely $A^{'} = S ∖ A$ .
intersection: The intersection of events $A, B$ denoted $A \cap B$ is the event of all outcomes that are in both $A$ and $B$ .
union: The union of two events $A, B$ denoted $A \cup B$ is the event consisting of all outcomes either in $A$ or $B$ .

Example

Continuing our coin flip example, and looking at coin flip events, we see:

$A \cup B = B = {T T T, H T T, T H T, T T H}$ in this specific case
$A \cap B = A = {H T T, T H T, T T H}$ in this specific case
$A^{'} = {H H H, H H T, H T H, T H H, T T T}$

We also have [De Morgan's Laws](8 College/82 Sophomore Year/823 Spring/CSC248-DiscreteStructures/Lecture Notes.pdf#page=12) that can be extended to sets (without proof here):

De Morgan's Laws

Let $A, B \in S$ be any two events. Then:

$(A \cup B)^{'} = A^{'} \cap B^{'}$
$(A \cap B)^{'} = A^{'} \cup B^{'}$

If $A, B$ have no events in common, they are disjoint or mutually exclusive events. We say $A \cap B = \emptyset$ when this happens, where $\emptyset$ is the empty set/event.

Venn Diagrams

We can use Venn Diagrams to show some events within our sample space:

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References

[[Matthew A. Carlton, Jay L. Devore - Probability with STEM Applications-Wiley (2020).pdf#page=39]]