Difference between revisions of "Distributive property"
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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>. | <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. |
| − | Also note that there is no particular reason that distributivity should be one-way, as it is with conventional multiplication and addition. For example, in a [[distributive lattice]], each of the operations [[meet]] and [[join]] distributes over the other. | + | 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>. |
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| + | (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 14:30, 23 October 2006
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.)
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