I have the following definition of an algebra:
An algebra of sets is a family $A$ of subsets of a certain set $X$, satisfying the following rules:
- $\emptyset, X \in A$
- If $a, b \in A$, then $a \cup b \in A$
- If $a, b \in A$, then $a\cap b \in A$
- If $a\in A$, then $a^c \in A$
So I understand that, ordinarily, the power set $P(X)$ of a finite set $X$ is an algebra, but I really struggle with extending things to infinite cases and I just can't figure out if the power set $P(\mathbb Z)$ (where $\mathbb Z$ is the set of integers) is an algebra. I think it must satisfy the first three rules but does it satisfy the last one? I just have no idea. Sorry for the stupid question, I promise I'm trying my best.
Yes, it satisfies the last one. The complement of a subset of $\Bbb{Z}$ is again a subset of $\Bbb{Z}$. In fact this has nothing to do with $\Bbb{Z}$; in this way the power set of any set is an algebra.