# What is meant by closed under complementation?

I was going through the probability and measure chapter of testing of hypothesis book by L.H. Lehman, where I found this "A class of sets that contains Z and is closed under complementation and countable unions is a σ-field". I was not able to understand closed under complementation in this matter. Any help would be appreciated.

Closed under complementation means that if a set $E\in\mathcal A$, where $\mathcal A$ is our $\sigma$-algebra, then we must have $E^c\in \mathcal A$ as well. Note that closed under countable unions and closed under complementation implies closed under countable intersection by De Morgan's laws.
In your case it means, if $A \in X$ then $A^c \in X$ for all $A \in X$.
• And Metin's notation used $A^c$ for the complement $Z \setminus A$. Presumably we have seen previously in Lehman that it is a class of subsets of $Z$. – GEdgar Mar 23 '13 at 16:03