Quick theory overview:
A collection of subsets \(\mathcal{F}\) of a sample space Ω is a Sigma Algebra if it satisfies three axioms:
- Axiom 1: It contains the empty set (∅) and the sample space (Ω).
- Axiom 2 (Closure under Complementation): If a set A is in 𝕄, its complement Ac must also be in 𝕄.
- Axiom 3 (Closure under Countable Unions): The union of any countable collection of sets in 𝕄 is also in 𝕄.
Procedure:
- A sample space Ω and an initial collection of sets are given.
- From the "Available Sets" panel, click to select the minimum number of additional sets needed to make the collection a valid Sigma Algebra.
- Selected sets will move to the "Your Added Sets" panel. You can click them again to de-select them.
- If you are stuck, press the "Get Hint" button.
- Once you are confident, click "Check Answer". The "Observations" panel will provide feedback.
- Press "New Problem" to try again with a different initial collection.