Distributivity: Difference between revisions

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* There are three closely connected examples where each of two operations distributes over the other:
* 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 [[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 [[propositional logic]], [[Conjunction (logical and)|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.
** In a [[distributive lattice]], [[join]] distributes over [[meet]] and meet distributes over join.[[Category:Suggestion Bot Tag]]

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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