Set Intersection Preserves Subsets/Families of Sets

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Theorem

Let $I$ be an indexing set.

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$.

Let:

$\forall \beta \in I: A_\beta \subseteq B_\beta$


Then:

$\ds \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in I} B_\alpha$


Corollary 1

Let $I$ be an indexing set.

Let $\family {B_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$.


Let $A$ be a set such that $A \subseteq B_\alpha$ for all $\alpha \in I$.


Then:

$\ds A \subseteq \bigcap_{\alpha \mathop \in I} B_\alpha$


Corollary 2

Let $I$ be an indexing set.

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$.

Let:

$\forall \beta \in I: A_\beta \subseteq B_\beta$


Then:

$\ds \bigcap_{\alpha \mathop \in I} B_\alpha = \O \implies \bigcap_{\alpha \mathop \in I} A_\alpha = \O$


Proof

\(\ds x\) \(\in\) \(\ds \bigcap_{\alpha \mathop \in I} A_\alpha\)
\(\ds \leadsto \ \ \) \(\ds \forall \alpha \in I: \, \) \(\ds x\) \(\in\) \(\ds A_\alpha\) Definition of Intersection of Family
\(\ds \leadsto \ \ \) \(\ds \forall \alpha \in I: \, \) \(\ds x\) \(\in\) \(\ds B_\alpha\) Definition of Subset
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \bigcap_{\alpha \mathop \in I} B_\alpha\) Definition of Intersection of Family

By definition of subset:

$\ds \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in I} B_\alpha$

$\blacksquare$


Sources

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