Properties of Algebras of Sets 1

Recall from the Algebras of Sets page that if $X$ is a set then an algebra on $X$ is a nonempty collection $\mathcal A$ of subsets of $X$ satisfying the following properties:

• For every $A_1, A_2 \in \mathcal A$ we have that $A_1 \cup A_2 \in \mathcal A$.

• For every $A \in \mathcal A$ we have that $A^c \in \mathcal A$.

We will now prove some nice properties of algebras of sets. More properties can be found on the Properties of Algebras of Sets 2 page.

Proposition 1: If $\mathcal A$ is an algebra on a set $X$ and if $A_1, A_2, ..., A_n \in \mathcal A$ for any $n \in \mathbb{N}$ then: a) $\displaystyle{\bigcup_{i=1}^{n} A_i \in \mathcal A}$. b) $\displaystyle{\bigcap_{i=1}^{n} A_i \in \mathcal A}$.

• Proof of a) Let $\mathcal A$ be an algebra on $X$. Assume that for $k \in \mathbb{N}$, $k \leq n$, that if $A_1, A_2, ..., A_{k-1}, A_k \in \mathcal A$ that $\displaystyle{\bigcup_{i=1}^{k-1} A_i \in \mathcal A}$. Then by property (1) of the definition we have that:

(1)

\begin{align} \quad \bigcup_{i=1}^{k} A_i = \left ( \bigcup_{i=1}^{k-1} A_i \right ) \cup A_k \in \mathcal A \end{align}

• By the principle of mathematical induction, for all $n \in \mathbb{N}$ we have that if $A_1, A_2, ..., A_n \in \mathcal A$ then $\displaystyle{\bigcup_{i=1}^{n} A_i \in \mathcal A}$. $\blacksquare$

• Proof of b) Let $\mathcal A$ be an algebra on $X$. Assume that for $k \in \mathbb{N}$, $k < n$, that if $A_1, A_2, ..., A_{k-1}, A_k \in \mathcal A$ that $\displaystyle{\bigcap_{i=1}^{k-1} A_i \in \mathcal A}$. Then by the previous proposition we have that:

(2)

\begin{align} \quad \bigcap_{i=1}^{k} A_i = \left ( \bigcap_{i=1}^{k-1} A_i \right ) \cap A_k \in \mathcal A \end{align}

• By the principle of mathematical induction, for all $n \in \mathbb{N}$ we have that if $A_1, A_2, ..., A_n \in \mathcal A$ then $\displaystyle{\bigcap_{i=1}^{n} A_i \in \mathcal A}$. $\blacksquare$