# Set Intersection Preserves Subsets/Families of Sets

## 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:

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

### Corollary 1

Let $I$ be an indexing set.

Let $\left \langle {B_\alpha} \right \rangle_{\alpha \mathop \in I}$ 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:

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

### 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:

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

## Proof

 $\displaystyle x$ $\in$ $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha$ Definition of Intersection of Family $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle B_\alpha$ Definition of Subset $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \bigcap_{\alpha \mathop \in I} B_\alpha$ Definition of Intersection of Family

By definition of subset:

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

$\blacksquare$