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In [[abstract algebra]] and [[logic]], the '''distributive property''' of [[binary operation]]s generalizes the '''distributive law''' from [[elementary algebra]]. In [[propositional calculus|propositional logic]], '''distribution''' refers to two [[validity|valid]] [[rule of replacement|rules of replacement]]. The rules allow one to reformulate [[logical conjunction|conjunctions]] and [[logical disjunction|disjunctions]] within [[formal proof|logical proofs]].  
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For example, in [[arithmetic]]:
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: 2 · (1 + 3) = (2 · 1) + (2 · 3), but 2 / (1 + 3) ≠ (2 / 1) + (2 / 3).


In the left-hand  side of the first equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the products added afterwards.
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Because these give the same final answer (8), we say that multiplication by 2 ''distributes'' over addition of 1 and 3.
Since we could have put any [[real number]]s in place of 2, 1, and 3 above, and still have obtained a true equation, we say that [[multiplication]] of real numbers ''distributes'' over [[addition]] of real numbers.


==Definition==
The special formula is due to the fact which when the particular metal is subjected in order to air, it types a protective oxide exterior in order to shield against rust. Many aluminum patio furniture is purchased with a powdered coating. All of this provides the aluminum an additional shield against scratches.


Given a [[Set (mathematics)|set]] ''S'' and two [[binary operator]]s <math>*</math> and <math>+</math> on ''S'', we say that the operation <math>*</math>
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* is ''left-distributive'' over <math>+</math> if, [[given any]] elements ''x'', ''y'', and ''z'' of ''S'',
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::<math>x * (y + z) = (x * y) + (x * z)</math>
* is ''right-distributive'' over <math>+</math> if, [[given any]] elements ''x'', ''y'', and ''z'' of ''S'':
::<math>(y + z) * x = (y * x) + (z * x)</math>
* is ''distributive'' over <math>+</math> if it is left- and right-distributive.<ref>Ayres, [http://books.google.com/books?id=7r3bVWx2GHkC&pg=PA20&dq=%22binary+operation%22+%22Left+distributive%22#PPA20,M1 p. 20].</ref>


Notice that when <math>*</math> is [[commutative]], then the three above conditions are [[logical equivalence|logically equivalent]].
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== Propositional logic ==
All commercial furniture, when selected for a specific contract, typically is thought to be contract furniture and even all the contract furniture bought for commercial purposes typically is considered commercial furniture. The line distinguishing them typically is really thin. One simple difference between the two typically is that the contract furniture is made for optimum durability and even to expect to have a longer lifespan.
{{Transformation rules}}
2.Design  Learning whether or not the style is general, custom and / or distinctive make we mindful of the actual real expense of this product. In a number of cases, distinctive designs is a large number of expensive and even general designs is least expensive. This is not constantly real and even this will be emphasized- particularly when operating straight with a custom furniture manufacturer. Bypassing suppliers and even operating straight with a manufacturer usually causes significant expense savings.
 
Speak with the owners of retail stores in order to allow them learn you're just inside the market for a real bargain. Ask them when their display furniture are upwards for grabs to make contact with you. They are searching for a bargain for you, not to mention saves you going back over and over again.
=== Rule of replacement ===
Modern furniture can be built with a complete selection brand new components consisting of laminated ply-woods, plastic, and even steel. The designs have in addition changed according to the necessity of space, cost, and even environment. Modern designs make use of straight lines and even components such as metal for workplace and even house furniture. Powder coated aluminum extrusions are in addition useful for compact furniture.
In standard truth-functional propositional logic, ''distribution''<ref>Moore and Parker</ref><ref>Copi and Cohen</ref><ref>Hurley</ref> are two valid rule of replacement. The rules allow one to distribute certain [[logical connective]]s within [[well-formed formula|logical expressions]] in logical proofs. The rules are:
 
:<math>(P \and (Q \or R)) \Leftrightarrow ((P \and Q) \or (P \and R))</math>
and
:<math>(P \or (Q \and R)) \Leftrightarrow ((P \or Q) \and (P \or R))</math>
 
where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a proof with."
 
=== Truth functional connectives ===
''Distributivity'' is a property of some logical connectives of truth-functional [[propositional logic]]. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional [[tautology (logic)|tautologies]].
 
'''Distribution of conjunction over conjunction'''
:<math>(P \and (Q \and R)) \leftrightarrow ((P \and Q) \and (P \and R))</math>
'''Distribution of conjunction over disjunction'''<ref>[[Bertrand Russell|Russell]] and [[Alfred Whitehead|Whitehead]], ''[[Principia Mathematica]]''</ref>
:<math>(P \and (Q \or R)) \leftrightarrow ((P \and Q) \or (P \and R))</math>
'''Distribution of disjunction over conjunction'''<ref>[[Bertrand Russell|Russell]] and [[Alfred Whitehead|Whitehead]], ''[[Principia Mathematica]]''</ref>
:<math>(P \or (Q \and R)) \leftrightarrow ((P \or Q) \and (P \or R))</math>
'''Distribution of disjunction over disjunction'''
:<math>(P \or (Q \or R)) \leftrightarrow ((P \or Q) \or (P \or R))</math>
'''Distribution of implication'''
:<math>(P \to (Q \to R)) \leftrightarrow ((P \to Q) \to (P \to R))</math>
'''Distribution of implication over equivalence'''
:<math>P \to (Q \leftrightarrow R) \leftrightarrow ((P \to Q) \leftrightarrow (P \to R))</math>
'''Distribution of disjunction over equivalence'''
:<math>(P \or (Q \leftrightarrow R)) \leftrightarrow ((P \or Q) \leftrightarrow (P \or R))</math>
'''Double distribution'''
:<math>\begin{align}
  ((P \and Q) \or (R \and S)) \leftrightarrow (((P \or R) \and (P \or S)) \and ((Q \or R) \and (Q \or S))) \\
  ((P \or Q) \and (R \or S)) \leftrightarrow (((P \and R) \or (P \and S)) \or ((Q \and R) \or (Q \and S)))
\end{align}</math>
 
==Examples==
 
# Multiplication of [[number]]s is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from [[natural number]]s to [[complex number]]s and [[cardinal number]]s.
# [[Ordinal arithmetic#Multiplication|Multiplication]] of [[ordinal number]]s, in contrast, is only left-distributive, not right-distributive.
# The [[cross product]] is left- and right-distributive over [[vector addition]], though not commutative.
# [[matrix (mathematics)|Matrix]] [[Matrix multiplication|multiplication]] is distributive over [[matrix addition]], though also not commutative.
# The [[union (set theory)|union]] of sets is distributive over [[intersection (set theory)|intersection]], and intersection is distributive over union.
# Logical disjunction ("or") is distributive over logical conjunction ("and"), and conjunction is distributive over disjunction.
# For [[real number]]s (and for any [[totally ordered set]]), the maximum operation is distributive over the minimum operation, and vice-versa: max(''a'',min(''b'',''c'')) = min(max(''a'',''b''),max(''a'',''c'')) and min(''a'',max(''b'',''c'')) = max(min(''a'',''b''),min(''a'',''c'')).
# For [[integer]]s, the [[greatest common divisor]] is distributive over the [[least common multiple]], and vice-versa: gcd(''a'',lcm(''b'',''c'')) = lcm(gcd(''a'',''b''),gcd(''a'',''c'')) and lcm(''a'',gcd(''b'',''c'')) = gcd(lcm(''a'',''b''),lcm(''a'',''c'')).
# For real numbers, addition distributes over the maximum operation, and also over the minimum operation: ''a'' + max(''b'',''c'') = max(''a''+''b'',''a''+''c'') and ''a'' + min(''b'',''c'') = min(''a''+''b'',''a''+''c'').
 
==Distributivity and rounding==
In practice, the distributive property of multiplication (and division) over addition may appear to be compromised or lost because of the limitations of [[arithmetic precision]].  For example, the identity ⅓ + ⅓ + ⅓ = (1 + 1 + 1) / 3  appears to fail if the addition is conducted in [[decimal arithmetic]]; however, if many [[significant digit]]s are used, the calculation will result in a closer approximation to the correct results.  For example, if the arithmetical calculation takes the form: 0.33333 + 0.33333 + 0.33333 = 0.99999 ≠ 1, this result is a closer approximation than if fewer significant digits had been used. Even when fractional numbers can be represented exactly in arithmetical form, errors will be introduced if those arithmetical values are rounded or truncated.  For example, buying two books, each priced at £14.99 before a [[VAT|tax]] of 17.5%, in two separate transactions will actually save £0.01, over buying them together: £14.99 × 1.175 = £17.61 to the nearest £0.01, giving a total expenditure of £35.22, but £29.98 × 1.175 = £35.23.  Methods such as [[Rounding#Round half to even|banker's rounding]] may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.
 
==Distributivity in rings==
Distributivity is most commonly found in [[ring (algebra)|ring]]s and [[distributive lattice]]s.
 
A ring has two binary operations (commonly called "+" and "*"), and one of the requirements of a ring is that * must distribute over +.
Most kinds of numbers (example 1) and matrices (example 4) form rings.
A [[lattice (order)|lattice]] is another kind of [[algebraic structure]] with two binary operations, ∧ and ∨.
If either of these operations (say ∧) distributes over the other (∨), then ∨ must also distribute over ∧, and the lattice is called distributive. See also the article on [[distributivity (order theory)]].
 
Examples 4 and 5 are [[Boolean algebra (structure)|Boolean algebra]]s, which can be interpreted either as a special kind of ring (a [[Boolean ring]]) or a special kind of distributive lattice (a [[Boolean lattice]]). Each interpretation is responsible for different distributive laws in the Boolean algebra. Examples 6 and 7 are distributive lattices which are not Boolean algebras.
 
Failure of one of the two distributive laws brings about [[near-ring]]s and [[near-field (mathematics)|near-field]]s instead of rings and [[division ring]]s respectively. The operations are usually configured to have the near-ring or near-field distributive on the right but not on the left.
 
Rings and distributive lattices are both special kinds of [[rig (algebra)|rig]]s, certain generalizations of rings.
Those numbers in example 1 that don't form rings at least form rigs.
[[Near-rig]]s are a further generalization of rigs that are left-distributive but not right-distributive; example 2 is a near-rig.
 
==Generalizations of distributivity==
 
In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially in [[order theory]] one finds numerous important variants of distributivity, some of which include infinitary operations, such as the [[infinite distributive law]]; others being defined in the presence of only ''one'' binary operation, such as the according definitions and their relations are given in the article [[distributivity (order theory)]]. This also includes the notion of a '''[[completely distributive lattice]]'''.
 
In the presence of an ordering relation, one can also weaken the above equalities by replacing = by either ≤ or ≥. Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of '''sub-distributivity''' as explained in the article on [[interval arithmetic]].
 
In [[category theory]], if ''(S, μ, η)'' and ''(S', μ', η')'' are [[monad (category theory)|monad]]s on a [[category (mathematics)|category]] ''C'', a '''distributive law''' ''S.S' → S'.S'' is a [[natural transformation]] ''λ : S.S' → S'.S'' such that (''S' '', λ) is a [[lax map of monads]] ''S → S'' and (''S'', λ) is a [[colax map of monads]] ''S' → S' ''. This is exactly the data needed to define a monad structure on ''S'.S'': the multiplication map is ''S'μ.μ'S².S'λS'' and the unit map is ''η'S.η''. See: [[distributive law between monads]].
 
A [[generalized distributive law]] has also been proposed in the area of [[information theory]].
 
==Notes==
{{reflist}}
 
==References==
* Ayres, Frank, ''Schaum's Outline of Modern Abstract Algebra'', McGraw-Hill; 1st edition (June 1, 1965). ISBN 0-07-002655-6.
 
== External links ==
{{Wiktionary|distributivity}}
*[http://www.cut-the-knot.org/Curriculum/Arithmetic/DistributiveLaw.shtml A demonstration of the Distributive Law] for integer arithmetic (from [[cut-the-knot]])
 
[[Category:Abstract algebra]]
[[Category:Binary operations|*Distributivity]]
[[Category:Elementary algebra]]
[[Category:Rules of inference]]
[[Category:Theorems in propositional logic]]

Revision as of 16:14, 20 February 2014

Consider regardless of whether you could pay for someone to fix or simply clean just about any difficulty with the show furniture. The ideal deals will be on furniture that has a lot of kind of damage, however in the event you are receiving a immense discount, then you may find it will be worth the when to go for the more damaged article.

As the name suggests, commercial furniture is actually designed as well as prepared for commercial settings such as offices, dining, lobbies etc. Contract furniture is actually manufactured and / or supplied for a specific contract it is a much broader term. Then again, for a layman the terms contract furniture as well as commercial furniture tend to be often utilized interchangeably. simply removals

Wood typically is not the best option with regards to a grassy area, only due to the wetness issue. However, in case wooden typically is just what you may be looking, then teak outdoor furniture is the greatest choice. It produces unique oils to help shield the surface from weather conditions scenarios or perhaps stains. Remember wooden definitely will break in cold temperatures, and so you could also need to shop the furniture in the winter.When searching for brand new furniture, you may find that the number one deals usually are virtually appropriate before your eye. A large number of furniture shops promote their show furniture at incredible prices- as well as being at the appropriate place at the appropriate time could imply slashing as much as 90 off the expenses of your furniture. Display models are set up for customers to see, touch, as well as feel the furniture options prior to them. Many of these pieces could consist of furniture for the whole home, such as sofas, beds, dressers, kitchen area tables, as well as much more. Commonly firms use showcases for a few months to a few months, then promote it at reduced costs to make means for the brand new showcases, passing on tremendous savings your means.

The special formula is due to the fact which when the particular metal is subjected in order to air, it types a protective oxide exterior in order to shield against rust. Many aluminum patio furniture is purchased with a powdered coating. All of this provides the aluminum an additional shield against scratches.

one.Material- Information about the actual material chosen in the actual manufacturing will offer an in-depth knowing of the actual toughness and enable you realize the actual real value of the actual piece.

Normally, a couple of pieces are constantly found about inventory in any retail shop marketing kashmiri furniture a couple of other items may need to be made with order. The items that are plentiful in the actual kashmiri walnut furniture category are solid wood space partitions, center tables, end tables, couch sets, roll-over table tops, roll-over cupboards, and so on.

To purchase furniture on the net almost all you have to be able to do can be search for your needs. You are able to choose to locate firm sites and search their products for the particular exact furniture you want or alternatively you can choose to be able to search your needs and go look at the different businesses that show up because a happen. You may be able to search for just your needs and in addition not have to talk to a salesperson that desires to sell you some thing that you don't want. It is a great deal easier for you to get the furniture on the net you want.

All commercial furniture, when selected for a specific contract, typically is thought to be contract furniture and even all the contract furniture bought for commercial purposes typically is considered commercial furniture. The line distinguishing them typically is really thin. One simple difference between the two typically is that the contract furniture is made for optimum durability and even to expect to have a longer lifespan. 2.Design Learning whether or not the style is general, custom and / or distinctive make we mindful of the actual real expense of this product. In a number of cases, distinctive designs is a large number of expensive and even general designs is least expensive. This is not constantly real and even this will be emphasized- particularly when operating straight with a custom furniture manufacturer. Bypassing suppliers and even operating straight with a manufacturer usually causes significant expense savings. Speak with the owners of retail stores in order to allow them learn you're just inside the market for a real bargain. Ask them when their display furniture are upwards for grabs to make contact with you. They are searching for a bargain for you, not to mention saves you going back over and over again. Modern furniture can be built with a complete selection brand new components consisting of laminated ply-woods, plastic, and even steel. The designs have in addition changed according to the necessity of space, cost, and even environment. Modern designs make use of straight lines and even components such as metal for workplace and even house furniture. Powder coated aluminum extrusions are in addition useful for compact furniture.