Difference between revisions of "Distributive property"

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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
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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>.   
 
 
<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, 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 $\times$ 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.

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.)

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