Definition of Sigma Algebra vs Algebra

Question

I have a simple questions concerning how $\sigma$-Algebra and Algebra are defined in my textbook. In it, it defines:

$\sigma$-Algebra is a class of sets $A$ st:

1. $\Omega \in A$ (omega being the entire space)
2. A is closed under set complements
3. A is closed under countable unions

And Algebra is defined as a class st:

1. $\Omega \in A$ (omega being the entire space)
2. A is closed under set Differences
3. A is closed under unions

My confusion comes from the set difference vs set complements part. It seems to me that closure under set difference is a stronger condition than closure under set complements. But the textbook seems to be treating a $\sigma$-Algebra as just an algebra that is closed under countable unions. Are set difference and complements equivalent due to the other conditions?

Answer 1

Any $\sigma$-algebra is closed under set differences.

This is because $A\backslash B= A\cap (B^c$).