Intersection Operation on Supersets of Subset is Closed

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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 $\mathscr S$
\((1):\quad\) \(\displaystyle \leadsto \ \ \) \(\displaystyle T\) \(\subseteq\) \(\displaystyle A \cap B\) 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
\((2):\quad\) \(\displaystyle \leadsto \ \ \) \(\displaystyle A \cap B\) \(\in\) \(\displaystyle \powerset S\) 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


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