Distributive Laws

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Theorem

Set Theory

Intersection Distributes over Union

Set intersection is distributive over set union:

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


Union Distributes over Intersection

Set union is distributive over set intersection:

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


Arithmetic

On all the number systems:

natural numbers $\N$
integers $\Z$
rational numbers $\Q$
real numbers $\R$
complex numbers $\C$

the operation of multiplication is distributive over addition:

$m \paren {n + p} = m n + m p$
$\paren {m + n} p = m p + n p$