Distributivity: Difference between revisions
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In [[algebra]], '''distributivity''' is a property of two [[binary operation]]s which generalises the relationship between [[addition]] and [[multiplication]] in [[elementary algebra]] known as "multiplying out". For these elementary operations it is also known as the '''distributive law''', expressed as | In [[algebra]], '''distributivity''' is a property of two [[binary operation]]s which generalises the relationship between [[addition]] and [[multiplication]] in [[elementary algebra]] known as "multiplying out". For these elementary operations it is also known as the '''distributive law''', expressed as | ||
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Formally, let <math>\otimes</math> and <math>\oplus</math> be binary operations on a set ''X''. We say that <math>\otimes</math> '''left distributes over''' <math>\oplus</math>, or is '''left distributive''', if | Formally, let <math>\otimes</math> and <math>\oplus</math> be binary operations on a set ''X''. We say that <math>\otimes</math> '''left distributes over''' <math>\oplus</math>, or is '''left distributive''', if | ||
:<math> a \otimes (b \oplus c) = (a \ | :<math> a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c) \,</math> | ||
and <math>\otimes</math> '''right distributes over''' <math>\oplus</math>, or is '''right distributive''', if | and <math>\otimes</math> '''right distributes over''' <math>\oplus</math>, or is '''right distributive''', if | ||
:<math>(b \oplus c) \otimes a = (b \ | :<math>(b \oplus c) \otimes a = (b \otimes a) \oplus (c \otimes a) . \,</math> | ||
The laws are of course equivalent if the operation <math>\otimes</math> is [[commutative]]. | The laws are of course equivalent if the operation <math>\otimes</math> is [[commutative]]. | ||
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* There are three closely connected examples where each of two operations distributes over the other: | * There are three closely connected examples where each of two operations distributes over the other: | ||
** In [[set theory]], [[intersection]] distributes over [[union]] and union distributes over intersection; | ** In [[set theory]], [[intersection]] distributes over [[union]] and union distributes over intersection; | ||
** In [[propositional logic]], [[conjunction]] (logical and) distributes over [[disjunction]] (logical or) and disjunction distributes over conjunction; | ** In [[propositional logic]], [[Conjunction (logical and)|conjunction]] (logical and) distributes over [[disjunction]] (logical or) and disjunction distributes over conjunction; | ||
** In a [[ | ** In a [[distributive lattice]], [[join]] distributes over [[meet]] and meet distributes over join.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:01, 7 August 2024
In algebra, distributivity is a property of two binary operations which generalises the relationship between addition and multiplication in elementary algebra known as "multiplying out". For these elementary operations it is also known as the distributive law, expressed as
Formally, let and be binary operations on a set X. We say that left distributes over , or is left distributive, if
and right distributes over , or is right distributive, if
The laws are of course equivalent if the operation is commutative.
Examples
- In a ring, the multiplication distributes over the addition.
- In a vector space, multiplication by scalars distributes over addition of vectors.
- There are three closely connected examples where each of two operations distributes over the other:
- In set theory, intersection distributes over union and union distributes over intersection;
- In propositional logic, conjunction (logical and) distributes over disjunction (logical or) and disjunction distributes over conjunction;
- In a distributive lattice, join distributes over meet and meet distributes over join.