# Difference between revisions of "Distributive property"

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Note that if the [[operation]] <math>\times</math> is [[commutative property | commutative]], these two conditions are the same, but if <math>\times</math> does not commute then we could have operations which ''left-distribute'' but do not ''right-distribute'', or vice-versa. | Note that if the [[operation]] <math>\times</math> is [[commutative property | commutative]], these two conditions are the same, but if <math>\times</math> does not commute then we could have operations which ''left-distribute'' but do not ''right-distribute'', or vice-versa. | ||

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+ | Key Note - This isn't an example of the Distributive Property! | ||

+ | <cmath>a(b \times c) = ab \times ac.</cmath> This is actually using the Associative Property, NOT THE DISTRIBUTIVE PROPERTY! | ||

Also note that there is no particular reason that distributivity should be one-way, as it is with conventional multiplication and addition. For example, the [[set]] operations [[union]] (<math>\cup</math>) and [[intersection]] (<math>\cap</math>) distribute over each other: for any sets <math>A, B, C</math> we have <math>A \cup (B \cap C) = (A \cup B) \cap (A \cup C)</math> and <math>A \cap(B \cup C) = (A \cap B) \cup (A \cap C)</math>. | Also note that there is no particular reason that distributivity should be one-way, as it is with conventional multiplication and addition. For example, the [[set]] operations [[union]] (<math>\cup</math>) and [[intersection]] (<math>\cap</math>) distribute over each other: for any sets <math>A, B, C</math> we have <math>A \cup (B \cap C) = (A \cup B) \cap (A \cup C)</math> and <math>A \cap(B \cup C) = (A \cap B) \cup (A \cap C)</math>. | ||

(In fact, this is a special case of a more general setting: in a [[distributive lattice]], each of the operations [[meet]] and [[join]] distributes over the other. Meet and join correspond to union and intersection when the lattice is a [[boolean lattice]].) | (In fact, this is a special case of a more general setting: in a [[distributive lattice]], each of the operations [[meet]] and [[join]] distributes over the other. Meet and join correspond to union and intersection when the lattice is a [[boolean lattice]].) | ||

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## Revision as of 09:11, 23 July 2020

Given two binary operations and acting on a set , we say that has the **distributive property** over (or *distributes over* ) if, for all we have and .

Note that if the operation is commutative, these two conditions are the same, but if does not commute then we could have operations which *left-distribute* but do not *right-distribute*, or vice-versa.

Key Note - This isn't an example of the Distributive Property! This is actually using the Associative Property, NOT THE DISTRIBUTIVE PROPERTY!

Also note that there is no particular reason that distributivity should be one-way, as it is with conventional multiplication and addition. For example, the set operations union () and intersection () distribute over each other: for any sets we have and .

(In fact, this is a special case of a more general setting: in a distributive lattice, each of the operations meet and join distributes over the other. Meet and join correspond to union and intersection when the lattice is a boolean lattice.)

*This article is a stub. Help us out by expanding it.*