# Distributivity

From Citizendium, the Citizens' Compendium

In algebra, **distributivity** is a property of two binary operations which generalises the relationship between addition and multiplication in elementary algebra known as "multiplying out". For these elementary operations it is also known as the **distributive law**, expressed as

Formally, let and be binary operations on a set *X*. We say that **left distributes over** , or is **left distributive**, if

and **right distributes over** , or is **right distributive**, if

The laws are of course equivalent if the operation is commutative.

## Examples

- In a ring, the multiplication distributes over the addition.
- In a vector space, multiplication by scalars distributes over addition of vectors.
- There are three closely connected examples where each of two operations distributes over the other:
- In set theory, intersection distributes over union and union distributes over intersection;
- In propositional logic, conjunction (logical and) distributes over disjunction (logical or) and disjunction distributes over conjunction;
- In a distributive lattice, join distributes over meet and meet distributes over join.