Intersection with Subset is Subset

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Theorem

$S \subseteq T \iff S \cap T = S$

where:

$S \subseteq T$ denotes that $S$ is a subset of $T$
$S \cap T$ denotes the intersection of $S$ and $T$.


Proof

Let $S \cap T = S$.

Then by the definition of set equality, $S \subseteq S \cap T$.

Thus:

\(\displaystyle S \cap T\) \(\subseteq\) \(\displaystyle T\) Intersection is Subset
\(\displaystyle \leadsto \ \ \) \(\displaystyle S\) \(\subseteq\) \(\displaystyle T\) Subset Relation is Transitive


Now let $S \subseteq T$.

From Intersection is Subset we have $S \supseteq S \cap T$.

We also have:

\(\displaystyle S\) \(\subseteq\) \(\displaystyle T\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle S \cap S\) \(\subseteq\) \(\displaystyle T \cap S\) Set Intersection Preserves Subsets
\(\displaystyle \leadsto \ \ \) \(\displaystyle S\) \(\subseteq\) \(\displaystyle S \cap T\) Intersection is Idempotent and Intersection is Commutative


So as we have:

\(\displaystyle S \cap T\) \(\subseteq\) \(\displaystyle S\)
\(\displaystyle S \cap T\) \(\supseteq\) \(\displaystyle S\)

it follows from the definition of set equality that:

$S \cap T = S$


So we have:

\(\displaystyle S \cap T = S\) \(\implies\) \(\displaystyle S \subseteq T\)
\(\displaystyle S \subseteq T\) \(\implies\) \(\displaystyle S \cap T = S\)

and so:

$S \subseteq T \iff S \cap T = S$

from the definition of equivalence.

$\blacksquare$


Also see


Sources