# Intersection Operation on Supersets of Subset is Closed

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## Contents

## 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.

$\blacksquare$

## Also see

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $5$: Subgroups: Exercise $1$