Difference between revisions of "Distributive property"

Latest revision as of 10:13, 23 July 2020

Given two binary operations $\times$ and $+$ acting on a set $S$ , we say that $\times$ has the distributive property over $+$ (or $\times$ distributes over $+$ ) if, for all $a, b, c \in S$ we have $a\times(b + c) = (a\times b) + (a \times c)$ and $(a + b) \times c = (a \times c) + (b \times c)$ .

Note that if the operation $\times$ is commutative, these two conditions are the same, but if $\times$ 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! $a(b \times c) = ab \times ac.$ 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 ( $\cup$ ) and intersection ( $\cap$ ) distribute over each other: for any sets $A, B, C$ we have $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap(B \cup C) = (A \cap B) \cup (A \cap C)$ .

(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.

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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
+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>.
-<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.
+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>.
+(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]].)
-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.
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Difference between revisions of "Distributive property"

Latest revision as of 10:13, 23 July 2020