# Find smallest $sigma$-algebra with intersections

Mathematics Asked by Puigi on December 17, 2020

I was asked to find the smallest $$sigma$$-algebra ($$sigma(Z)$$) generated by the following set $$Z$$={{1,2},{2,3},{3,4}} where the sample space is $$Omega$$={1,2,3,4}. The thing is that the middle element of $$Z$$ is causing me a lot of trouble. The smallest $$sigma(Z)$$ I could find was the following:

$$sigma(Z)$$={$$emptyset$$,$$Omega$$,{1},{2},{3},{4},{1,2},{2,3},{3,4},{1,3},{1,4},{2,4},{1,2,3},{2,3,4},{1,3,4},{1,2,4}}

I know that the $$sigma$$-algebra has to be closed under complements and unions, and that is how I got to that set. Is this right? Is there a faster way to solve this kind of problems?

The moment you get $${1},{2},{3},{4}$$ in the sigma algebra you can conclude that all subsets of $${1,2,3,4}$$ are also in it. (Because any subset of $${1,2,3,4}$$ is a finite union of singletons). In this case $${1}={1,2,}setminus {2,3}$$, $${2}={1,2}cap {2,3}$$, $${3}={2,3,}setminus ({1,2}$$ and $${4}={3,4}setminus ({2,3}$$.

Correct answer by Kavi Rama Murthy on December 17, 2020

There is a faster way : Since $${1},{2},{3},{4}in sigma (Z)$$, then $$sigma (Z)$$ is the power set of $${1,2,3,4},$$ and what you found is correct.

Answered by Surb on December 17, 2020

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