Distributive Laws

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Theorem

Intersection Distributes over Union

Set intersection is distributive over set union:

$R \cap \paren {S \cup T} = \paren {R \cap S} \cup \paren {R \cap T}$


Family of Sets

Let $I$ be an indexing set.

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

Let $B \subseteq S$.


Then:

$\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$

where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.


General Result

Let $S$ and $T$ be sets.

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

Let $\mathbb T$ be a subset of $\mathcal P \left({T}\right)$.


Then:

$\displaystyle S \cap \bigcup \mathbb T = \bigcup_{X \mathop \in \mathbb T} \left({S \cap X}\right)$


Union Distributes over Intersection

Set union is distributive over set intersection:

$R \cup \paren {S \cap T} = \paren {R \cup S} \cap \paren {R \cup T}$


Family of Sets

Let $I$ be an indexing set.

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

Let $B \subseteq S$.


Then:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$

where $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersection of $\family {A_\alpha}_{\alpha \mathop \in I}$.


General Result

Let $S$ and $T$ be sets.

Let $\powerset T$ be the power set of $T$.

Let $\mathbb T$ be a subset of $\powerset T$.


Then:

$\displaystyle S \cup \bigcap \mathbb T = \bigcap_{X \mathop \in \mathbb T} \paren {S \cup X}$


Examples

Example: $A \cap B \cap \paren {C \cup D} \subseteq \paren {A \cap D} \cup \paren {B \cap C}$

Let:

$P = A \cap B \cap \paren {C \cup D}$
$Q = \paren {A \cap D} \cup \paren {B \cap C}$

Then:

$P \subseteq Q$


Sources