# Intersection Operation on Supersets of Subset is Closed

## Theorem

Let $S$ be a set.

Let $T \subseteq S$ be a given subset of $S$.

Let $\powerset S$ denote the power set of $S$

Let $\mathscr S$ be the subset of $\powerset S$ defined as:

$\mathscr S = \set {Y \in \powerset S: T \subseteq Y}$

Then the algebraic structure $\struct {\mathscr S, \cap}$ is closed.

## Proof

Let $A, B \in \mathscr S$.

We have that:

 $\displaystyle T$ $\subseteq$ $\displaystyle A$ Definition of $\mathscr S$ $\displaystyle T$ $\subseteq$ $\displaystyle B$ Definition of $\mathscr S$ $\text {(1)}: \quad$ $\displaystyle \leadsto \ \$ $\displaystyle T$ $\subseteq$ $\displaystyle A \cap B$ Intersection is Largest Subset

and:

 $\displaystyle A$ $\subseteq$ $\displaystyle S$ Definition of Power Set $\displaystyle B$ $\subseteq$ $\displaystyle S$ Definition of Power Set $\displaystyle \leadsto \ \$ $\displaystyle A \cap B$ $\subseteq$ $\displaystyle S$ Intersection is Subset $\text {(2)}: \quad$ $\displaystyle \leadsto \ \$ $\displaystyle A \cap B$ $\in$ $\displaystyle \powerset S$ Definition of Power Set

Thus we have:

 $\displaystyle T$ $\subseteq$ $\displaystyle A \cap B$ from $(1)$ $\displaystyle A \cap B$ $\in$ $\displaystyle \powerset S$ from $(2)$ $\displaystyle \leadsto \ \$ $\displaystyle A \cap B$ $\in$ $\displaystyle \mathscr S$ Definition of $\mathscr S$

Hence the result by definition of closed algebraic structure.

$\blacksquare$