Sigma algebra generated by a piecewise function

[ellese]

I have this homework question and I'm a little bit stuck.

The question is:
Let (Ω, F , P ) be the unit interval probability space, and X be the random variable
defined by

(See attatchment)

Find the σ-algebra generated by X.

From my understanding of sigma-algebras I want to look for the set that is non-empty, closed under complementation and closed under countable unions.

The idea I have so far is to include the sets {1}{3} and {(3, 7]} but I'm not sure how to combine these into a sigma algebra. Do I want σ(X) = {∅, {1}{3}{(3, 7]}{[3,7]}, {1, (3,7]}, {1, 3} , {1, [3,7]}}?

 

[lippyka]

Any help would be much appreciated. Thank you! Yes, your understanding is correct. The sigma algebra generated by X is indeed $$\sigma (X) =\{\emptyset , \{1\}, \{3\}, \{7\}, \{1,3\}, \{3,7\}, \{1,7\}, \{1,3,7\}\}.$$