Difference between revisions of "Distributive Property"
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| − | + | The '''Distributive Property''' is a [[mathematic]]al [[property]] for two specific [[binary operation]]s. This name is also commonly used to refer to the distributive property of multiplication over addition. | |
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| + | == Statement == | ||
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| + | 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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| + | 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. | ||
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| + | Note that there is no particular reason that distributivity should always 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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| + | == See Also == | ||
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| + | * [[Algebra]] | ||
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| + | {{stub}} | ||
Latest revision as of 18:33, 17 September 2025
The Distributive Property is a mathematical property for two specific binary operations. This name is also commonly used to refer to the distributive property of multiplication over addition.
Statement
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.
Note that there is no particular reason that distributivity should always 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.)
See Also
This article is a stub. Help us out by expanding it.
