# Basic coin toss Borel sigma algebra question

The sample space $\Omega = \{ HH, HT, TH, TT \}$. What is the smallest $\sigma$ algebra containing the events $\{HH \},\{HT \}, \{TT \}, \{TH \}$?

I am having trouble visualizing this $\sigma$ algebra. When I consider taking the union of any of the events, I simply get an absurd answer like $\{ HHTT\}$. This of course makes no sense. How do I even define $\{HH \} \cup \{TT\}$??

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For your example, $\{HH \} \cup\{TT\}$ is the union of two sets, each of which contains a single outcome. The set $\{HH \}$ contains the outcome $HH$, and the set $\{TT \}$ contains the outcome $TT$. The union $\{HH \} \cup\{TT\}$ contains both $HH$ and $TT$, and it is written $\{HH,TT\}$ with a comma to distinguish between the two elements.

Similarly, the notation $\Omega = \{ HH, HT, TH, TT \}$ means that $\Omega$ is a set containing four outcomes.

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Take the empty set and $\Omega$ itself. Then the complements of each event ( eg complement of $\{HH\}$ is $\{HT, TH, TT\}$-which is an event), and unions (eg union of $\{HH\}$ and $\{TT\}$ is $\{HH, TT\}$ -event is one of these happens). Finally check that the number of elements you have written makes sense, ie that all the sets are there so that this agrees with the definition.

Letting $a=HH$, $b=TT$, $c=HT$, $d=TH$, the answer is the power set of this $2^4=16$ elements in total

$\emptyset$, $\Omega$,

$\{a\}$, $\{b\}$, $\{c\}$, $\{d\}$,

$\{a,b\}$, {a,c}, $\{a,d\}$, $\{c,d\}$, $\{b,d\}$, $\{b,c\}$

$\{a,b,c\}$, $\{a,b,d\}$, $\{a,c,d\}$, $\{b,c,d\}$.

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