# Difference between revisions of "Distributive property"

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− | Given two [[binary operation]]s | + | Given two [[binary operation]]s <math>\times</math> and <math>+</math> acting on a set <math>S</math>, we say that <math>\times</math> has the '''distributive property''' over <math>+</math> (or <math>\times</math> ''distributes over'' <math>+</math>) if, for all <math>a, b, c \in S</math> we have <math>a\times(b + c) = (a\times b) + (a \times c)</math> and <math>(a + b) \times c = (a \times c) + (b \times c)</math>. |

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− | <math>a\times(b + c) = (a\times b) + (a \times c)</math> and <math>(a + b) \times c = (a \times c) + (b \times c)</math>. | ||

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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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 21:48, 20 April 2008

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.

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.*