Elementary event in event space I encountered a very basic question of probability. 
Consider the sample space Ω = {a,b,c,d} and assume that the only elementary events in the Event space F defined on Ω are {a} and {b}. Explicitly enumerate all the events in F.
What does the phrase "elementary events in the event space" mean? 
 A: Let us follow the definitions of Wikipedia. An elementary event is an event which contains only a single outcome in the sample space; which is the set of all possible outcomes of an experiment. Looking at the definition of a probability space we have that the event space $\mathcal{F}$ has to be a sigma algebra on the sample space $\Omega$.
Now we can explicit the elements in $\mathcal{F}$. By definition $\emptyset,\{a\},\{b\} \in \mathcal{F}$. Taking complements also $\{a,b,c,d\}, \{b,c,d\}, \{a,c,d\} \in\mathcal{F}$, and considering the union, and the complement again, also $\{a,b\},\{c,d\}\in\mathcal{F}$.
Now we claim that no other subset of $\Omega$ lies in $\mathcal{F}$, clearly no other triple can do that, otherwise, taking the complement, we would have elementary events in $\mathcal{F}$ different from $\{a\}$ and $\{b\}$, and smilarly we cannot have the cited elementary events. We are left with the following subsets of $\Omega$ with two elements: $\{a,c\},\{a,d\},\{b,c\},\{b,d\}$ but they are not in $\mathcal{F}$, otherwise intersecting with $ \{b,c,d\}, \{a,c,d\}$ we would have one of the prohibited elementary events.
Summing up:
$$\mathcal{F} = \{\emptyset,\{a\},\{b\} , \{a,b\},\{c,d\}, \{b,c,d\}, \{a,c,d\},\{a,b,c,d\} \}.$$
