Quick theory overview:
A collection of events, as a subcollection \(\mathcal{F}\) of the set of all subsets of \(\Omega\), which satisfy the following properties
If \(A \; \& \; B \in \mathcal{F}\), then \(A \cup B \in \mathcal{F}\)
If \(A \in \mathcal{F}\), then \(A^{C} \in \mathcal{F}\)
\(\phi \in \mathcal{F}\)
is called as a Sigma Algebra.
Procedure:
You are give a sample space \(\Omega\) and a collection C containing some events
from the sample space.
You have to select the minimum number of subsets of S that should be added to C for it to become a
sigma algebra over \(\Omega\).
You can add the events from the given list by clicking on it in the top panel.
You can remove the events from the collection C by clicking on it in the bottom
panel.
Once you have selected the events, click on the submit button to check your answer.
Given the sample space \(\Omega\) = \(\{a,b,c,d\}\) and a collection \(C = \{\{\},\{a,b,c,d\},\{a\}\}\), which of
the following events should be added to \(C\) for it to become a sigma algebra over \(\Omega\).
\(\{
b\}\)
\(\{
c\}\)
\(\{
d\}\)
\(\{
ab\}\)
\(\{
ac\}\)
\(\{
ad\}\)
\(\{
bc\}\)
\(\{
bd\}\)
\(\{
cd\}\)
\(\{
abc\}\)
\(\{
abd\}\)
\(\{
acd\}\)
\(\{
bcd\}\)
\(\{
b\}\)
\(\{
c\}\)
\(\{
d\}\)
\(\{
ab\}\)
\(\{
ac\}\)
\(\{
ad\}\)
\(\{
bc\}\)
\(\{
bd\}\)
\(\{
cd\}\)
\(\{
abc\}\)
\(\{
abd\}\)
\(\{
acd\}\)
\(\{
bcd\}\)
Observations
Given the sample space \(\Omega\) = {a,b,c,d} and a collection C = {{},{a,b,c,d},{a},{b}}, which
of the following events should be added to C for it to become a sigma algebra over \(\Omega\).