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	AllRefer.com - associative law (Mathematics) - Encyclopedia (Site not responding. Last check: 2007-10-07)
		associative law, in mathematics, law holding that for a given operation combining three quantities, two at a time, the initial pairing is arbitrary; e.g., using the operation of addition, the numbers 2, 3, and 4 may be combined (2+3)+4=5+4=9 or 2+(3+4)=2+7=9.
		More generally, in addition, for any three numbers a, b, and c the associative law is expressed as (a+b)+c=a+(b+c).
		When an operation is associative, the parentheses indicating which quantities are first to be combined may be omitted, e.g., (2+3)+4=2+(3+4)=2+3+4.
	reference.allrefer.com /encyclopedia/A/assoc-law.html (223 words)

	Associativity - Wikipedia, the free encyclopedia
		This article is about associativity in mathematics, for associativity in central processor unit memory cache architecture, see CPU cache.
		Associative operations are abundant in mathematics, and in fact most algebraic structures explicitly require their binary operations to be associative.
		Power associativity and alternativity are weak forms of associativity.
	en.wikipedia.org /wiki/Associativity (616 words)

	[No title]
		The informal intuition behind associativity (the property of being associative) is that if one has an expression in which there are many parentheses and the only operation performed in this expression is that defined by an associative law of composition, then one may ignore the parentheses.
		An example of an associative law of composition is addition on the integers, Z.
		Here we use the same multiplicative notation for the laws of composition of G_1 and G_2, even though there is no requirement that their laws of composition be the same.
	www.math.harvard.edu /~knill/sofia/data/group.txt (4457 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		(Associative Law) If a, b, c are integers, then (a + b) + c = a + (b + c), and (a b) c = a (b c).
		(Law of Identity Elements) For all integer a, a + 0 = 0 + a = a; a • 1 = 1 • a = a.
		(Law of Additive Inverse) For all integer a, a + (—a) = (—a) + a = 0.
	longwood.cs.ucf.edu /courses/cot3100.fall00/section1/chap1defs.doc (550 words)

	associative law -- Britannica Concise Encyclopedia - The online encyclopedia you can trust!
		in mathematics, either of two laws relating to number operations of addition and multiplication, stated symbolically: a + (b + c) = (a + b) + c, and a(bc) = (ab)c; that is, the terms or factors may be associated in any way desired.
		While associativity holds for positive or negative, integral or fractional, rational or irrational, or real or imaginary numbers, there are...
		Natural law has been recognized since the ancient world to be a general body of rules of right conduct and justice common to all mankind.
	www.britannica.com /ebc/article-9355994?tocId=9355994 (893 words)

	All Elementary Mathematics - Study Guide - Arithmetics - Laws of addition and multiplication...
		Associative law of addition: (m + n) + k = m + (n + k) = m + n + k.
		Associative law of multiplication: (m · n) · k = m · (n · k) = m · n · k.
		This law expands the rules of operations with brackets (see the previous section).
	www.bymath.com /studyguide/ari/ari4.html (146 words)

	[No title]
		Example 1: Simplify 4 (6x) Solution: 4 (6x) = 4 * (6 * x) Clarify what each operation is. = (4 * 6) * x Use the associative law of multiplication, since the operation inside the parentheses was multiplication.
		The distributive law does not apply in Example 1, because the distributive law is used to multiply the sum or difference of two numbers by a third number.
		The associative law is appropriate for Example 1 because it tells us about the result of multiplying three quantities.
	www.austin.cc.tx.us /mparker/0330/eqn/eq4.doc (1897 words)

	Units in Physics Calculations for Physics 251 - R. D. Piccard (Site not responding. Last check: 2007-10-07)
		Here you will use the associative law, factoring out the units, which must be the same in order to do that.
		This is consistent with the idea that it makes no sense to add things that are of different kinds (a time plus a distance cannot yield an interesting sum!), and that if the same kind of thing is measured in different units, one or both must be converted so that the factoring-out can proceed.
		Remember, when adding two things of the same kind that have been measured in different units, you must first convert the units to match, and then you will be able to use the associative law to factor out the common units and add the numbers.
	oak.cats.ohiou.edu /~piccard/phys251/units.html (494 words)

	MySQL Lists: bugs: Re: Bug De-Morgan-Rules
		Commutative Law: For "numeric" operations: i) a + b = b + a ii) a * b = b * a i.e.
		Associative Law: For "numeric" operations: i) a + b + c = a + (b+c) = (a+b)+c ii) a * b * c = a * (bc) = (ab)*c i.e.
		Ternary algebra (as any other algebra) must support some kind of a commutative law, associative law and distributive law.
	lists.mysql.com /bugs/12892 (989 words)

	Boolean Functions
		The associative law of addition of three variables is expressed as
		The associative law of multiplication of three variables is expressed as
		A (B+C) = AB + AC This law states that ORing several variables and ANDing the result is equivalent of ANDing the single variable with each of the variables in the grouping, then ORing the result.
	scitec.uwichill.edu.bb /cmp/online/P10F/boolean.htm (268 words)

	Mathematics Magazine: June 2001
		An overview of the many different treatments of the General Associative Law in the literature is given.
		Then the following generalization of the General Associative Law is presented: A groupoid G (set with a single binary operation) is n-associative if the product of any n elements is independent of how they are associated, that is, if a1a2.
		The “Generalized General Associative Law” asserts that if n> 3 and G is n-associative, then G is (n+1)-associative.
	www.maa.org /pubs/mag_jun01_toc.html (1071 words)

	1.7 Class Notes
		Example 3 Use the commutative and/or associative laws of addition to write at least two expressions equivalent to
		Example 4 Use the commutative and/or associative laws of multiplication to rewrite 3(y5) as 15y.
		To factor an expression we reverse the statement of the distributive law.
	mtsu32.mtsu.edu:11197 /1_7_class_notes.htm (276 words)

	Math Forum - Ask Dr. Math
		Date: 02/12/99 at 11:37:56 From: Rose Zagaja Subject: Laws of Arithmetic I'm looking for information (definitions) on various laws of arithmetic - in particular, the distributive law, and also anything on the associative law, identity, and commutative equations.
		Date: 02/22/99 at 01:02:16 From: Doctor Bonnie Subject: Re: Laws of Arithmetic It is really nice to hear a student ask this particular question.
		For 2a and 3a, we say that F is closed with respect to addition and multiplication (respectively).
	mathforum.org /dr.math/problems/rose.02.12.99.html (317 words)

	Vectors
		The reason for this introduction to vectors is that many concepts in science, for example, displacement, velocity, force, acceleration, have a size or magnitude, but also they have associated with them the idea of a direction.
		The verification of the Associative law is shown in Panel 6.
		Associative Law for Multiplication: (m + n)A = mA + nA, where m and n are two different scalars.
	www.physics.uoguelph.ca /tutorials/vectors/vectors.html (1708 words)

	Search Results for associative - Encyclopædia Britannica
		in mathematics, either of two laws relating to number operations of addition and multiplication, stated symbolically: a + (b + c) = (a + b) + c, and a(bc) = (ab)c; that is, the terms or factors may...
		in mathematics, set that has a multiplication that is associative [a(bc) = (ab)c for any a, b, c] and that has an identity element and inverses for all elements of the set.
		in mathematics, the law relating to number operations stated symbolically, a(b + c + d) = ab + ac + ad; that is, the monomial factor a is distributed, or separately applied, to each term of the...
	www.britannica.com /search?ref=B04201&query=associative&submit=Find (517 words)

	Abstract groups
		Associative law: if x, y, z are in G then x(yz) = (xy)z.
		A group is defined by the law of composition of its members.
		The associative law is put in a slightly strange way.
	www-history.mcs.st-and.ac.uk /PrintHT/Abstract_groups.html (1884 words)

	[No title]
		In order for you to prove associativity of your suggested group law you have to use the relation y^2=x^3+ax+b which means that x and y may not be assumed to be algebraically independent.
		However, making it equivalent to a rational group law would in particular mean that the function field k(x)[y]/(y^2-x^3-ax-b) would be isomorphic to k(t) and this is known to be false.
		In fact, if we stick to an algebraically closed field then the assumption that we have a rational group law is precisely what excludes things like elliptic curves and forces the algebraic group to be a matrix group.
	www.math.niu.edu /~rusin/known-math/93_back/rational (1270 words)

	Abstract groups
		He then supposes that any two of the operations are distinct in the sense that no two produce the same effect.
		He follows Cayley in requiring closure, the associative law, and inverses.
		This appears to be a separate development by Kronecker who does not tie it in with previous work on groups.
	www-history.mcs.st-andrews.ac.uk /history/HistTopics/Abstract_groups.html (1893 words)

	associative law on Encyclopedia.com
		More generally, in addition, for any three numbers a, b, and c the associative law is expressed as (a + b)+ c = a +(b + c).
		Multiplication of numbers is also associative, i.e., (a × b)× c = a ×(b × c).
		In general, any binary operation, symbolized by [symbol], joining mathematical entities A, B, and C obeys the associative law if (A [symbol] B)[symbol] C = A [symbol](B [symbol] C) for all possible choices of A, B, and C.
	www.encyclopedia.com /html/a1/assoc-law.asp (382 words)

	Slide 1
		A + B = B + A (9) - this is known as the commutative law (OR)
		A B = B A (10) - this is known as the commutative law (AND)
		- this is known as the associative law (OR)
	www.ee.surrey.ac.uk /Personal/D.Carey/teaching/cs154/Lectures1-4_files/slide0012.htm (116 words)

	Math 1010 on-line - Numbers
		These laws are true for all and any natural numbers a, b, c.
		Actually they hold for all numbers we will encounter, but the whole point of building the number system is that we do this in such a way that the above rules remain true.
		The distributive law connects multiplication and addition and is the most crucial, and the most misused and misunderstood law in the above list.
	www.math.utah.edu /online/1010/numbers (901 words)

	ALGEBRAIC LAWS
		This chapter covers the laws used for solving algebraic equations.
		laws that govern the use of real numbers.
		These laws will be stated in written form as
	www.tpub.com /content/doe/h1014v1/css/h1014v1_115.htm (149 words)

	Multiplication of Polynomials-Chapter 10 Section 2 (Site not responding. Last check: 2007-10-07)
		We now will learn how to multiply polynomials employing techniques based, for the most part,on the distributive law, but also on the associative and commutative laws as well.
		That is the monomial must be multiplied times each term in the binomial and then combine and collect and simplify the expression.
		When an expression inside parentheses is raised to a power, the inside expression is the base.
	www.rose.edu /faculty/gjackson/preal/ch10.2.htm (653 words)

	Dehornoy's structure groups of algebraic identities (Site not responding. Last check: 2007-10-07)
		Abstract: Among the Thompson groups, F has been known to arrive from considerations of the associative law and V has been known to arrive from considerations of the associative and commuative laws.
		His construction agrees with the first sentence in that his construction leads to F when the set of identities consists of the associative law and leads to V when the set of identities consists of the associative law with the commuative law.
		His construction allows for introduction of "braided" versions of the commutative law.
	www.math.binghamton.edu /dept/topsem/04Abstracts/matt.html (116 words)

	The Field Properties (Site not responding. Last check: 2007-10-07)
		(a + b) + c = a + (b + c) Associative law for addition
		(a ´ b) ´ c = a ´ (b ´ c) Associative law for
		a ´ (b + c) = a ´ b + a ´ c and Distributive law
	www.faculty.sfasu.edu /cproctor/sect33.html (218 words)

	MATH 240: Definition of a Vector Space (Site not responding. Last check: 2007-10-07)
		A0 Closure: For all vectors v, w in V, the sum v + w belongs to V. Associative law: For all vectors u, v, w in V, (u + v) + w = u + (v + w).
		A2 Commutative law: For all vectors v, w in V, v + w = w + v.
		S3 Associative law: For all real numbers r,s and all vectors v in V, r
	www.math.niu.edu /~kholland/240/vectorspace.html (283 words)

	associative law
		Preexposure effects in spatial learning: from gestaltic to associative and attentional cognitive maps.
		The unwarranted pessimism of the new behavioral analysis of law.
		Let's not dam(n) the courts with more damage litigation: a comment on Frederickson's "Recovery for False Advertising...." (Paul D. Frederickson, American Business Law Journal, vol.
	www.infoplease.com /ce6/sci/A0805085.html (222 words)

	Distributive Law for Complex Numbers - Vector and Complex Numbers:
		We will be using rectangular coordinates [a, b] during most of this derivation (a cosmetic choice) in place of writing a + bi.
		Axioms for real numbers imply addition of points in the plane is associative and commutative.
		Then the associative law for addition follows not by an appeal to the associative law for real numbers, but from the associativity of head-to-tail addition of vectors and a delicate discussion of equality for vectors before and during addition operations.
	whyslopes.com /etc/ComplexNumbers/distributiveLaw.html (1634 words)

	PROPERTIES OF VECTOR ADDITION-COMMUTATIVE LAW OF VECTOR ADDITION-ASSOCIATIVE LAW OF VECTOR ADDITION
		This fact is referred to as the commutative law of vectr addition
		OF The law states that the sum of vectors remains same irrespective of their order or grouping in which they are arranged.
		This fact is known as the ASSOCIATIVE LAW OF VECTOR ADDITION.
	www.citycollegiate.com /vectorXIg.htm (102 words)

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