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