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 10:11, 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!
![\[a(b \times c) = ab \times ac.\]](http://latex.artofproblemsolving.com/2/f/0/2f0d80651a243d1d3144f2b6da755ea3ebad3fc8.png)
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.)
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