Intersection is Subset

From ProofWiki
Jump to: navigation, search

Theorem

The intersection of two sets is a subset of each:

$S \cap T \subseteq S$
$S \cap T \subseteq T$


General Result

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.


Then:

$\displaystyle \forall T \in \mathbb S: \bigcap \mathbb S \subseteq T$


Family of Sets

In the context of a family of sets, the result can be presented as follows:


Let $\left \langle{S_\alpha}\right \rangle_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.


Then:

$\displaystyle \forall \beta \in I: \bigcap_{\alpha \mathop \in I} S_\alpha \subseteq S_\beta$

where $\displaystyle \bigcap_{\alpha \mathop \in I} S_\alpha$ is the intersection of $\left \langle{S_\alpha}\right \rangle_{\alpha \mathop \in I}$.


Proof

\(\displaystyle x\) \(\in\) \(\displaystyle S \cap T\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle S \land x \in T\) $\quad$ Definition of Set Intersection $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle S\) $\quad$ Rule of Simplification $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle S \cap T\) \(\subseteq\) \(\displaystyle S\) $\quad$ Definition of Subset $\quad$


Similarly for $T$.

$\blacksquare$


Sources