# Properties of Algebras of Sets

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## Theorem

Let $X$ be a set.

Let $\mathfrak A$ be an algebra of sets on $X$.

Then the following hold:

- $(1): \quad$ The intersection of two sets in $\mathfrak A$ is in $\mathfrak A$.
- $(2): \quad$ The difference of two sets in $\mathfrak A$ is in $\mathfrak A$.
- $(3): \quad$ $X \in \mathfrak A$.
- $(4): \quad$ The empty set $\O$ is in $\mathfrak A$.

## Proof

Let:

- $X$ be a set
- $\mathfrak A$ be an algebra of sets on $X$
- $A, B \in \mathfrak A$

By the definition of algebra of sets, we have that:

- $A \cup B \in \mathfrak A$
- $\relcomp X A \in \mathfrak A$

Thus:

\(\displaystyle \) | \(\) | \(\displaystyle A, B \in \mathfrak A\) | |||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \relcomp X A \cup \relcomp X B \in \mathfrak A\) | Definition of Algebra of Sets | ||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \relcomp X {A \cap B} \in \mathfrak A\) | De Morgan's Laws: Complement of Intersection | ||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle A \cap B \in \mathfrak A\) | Definition of Algebra of Sets |

and so we have that the intersection of two sets in $\mathfrak A$ is in $\mathfrak A$.

Next:

\(\displaystyle \) | \(\) | \(\displaystyle A, B \in \mathfrak A\) | |||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle A \cap \relcomp X B \in \mathfrak A\) | from above | ||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle A \setminus B \in \mathfrak A\) | Set Difference as Intersection with Relative Complement |

and so we have that the difference of two sets in $\mathfrak A$ is in $\mathfrak A$.

We have that $\mathfrak A \ne \O$ and so $\exists A \subseteq X: A \in \mathfrak A$.

Then:

\(\displaystyle \relcomp X A\) | \(\in\) | \(\displaystyle \mathfrak A\) | Definition of Algebra of Sets | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \relcomp X A \cup A\) | \(\in\) | \(\displaystyle \mathfrak A\) | Definition of Algebra of Sets | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle X\) | \(\in\) | \(\displaystyle \mathfrak A\) | Union with Relative Complement |

Also, $\relcomp X A \cap A \in \mathfrak A$ from above.

So by Intersection with Relative Complement is Empty:

- $\O \in \mathfrak A$

$\blacksquare$