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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.

Kenny Wong
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isawahatonce
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  • Should the last one be: If $a\in A$, then $X\setminus a\in A$? – Mathematician 42 May 04 '17 at 08:37
  • I think the answer is yes. The complement of any subset of $\mathbb Z$ is also a subset of $\mathbb Z$. Whether the sets involved are infinite is not important. – Kenny Wong May 04 '17 at 08:37
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    In fact, with the above definition, the power set of any set is an algebra of sets. – Mathematician 42 May 04 '17 at 08:38
  • A more interesting algebra of sets is the finite-cofinite algebra, i.e. given a set $X$, you can consider finite subsets of $X$ and their complements. Also, $\sigma$-algebras are a special kind of algebra of sets. They appear naturally in measure theory and probability theory. – Mathematician 42 May 04 '17 at 08:40

1 Answers1

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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.

Servaes
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  • Don't we need to take "closed under complements" with respect to the original set though? (To say in this way the power set of any set is an algebra.) – independentvariable Dec 03 '21 at 22:31