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Associativity - Wikipedia, the free encyclopedia

Associative operations are abundant in mathematics, and in fact most algebraic ... In arithmetic, addition and multiplication of real numbers are associative; i.e. ...
http://en.wikipedia.org/wiki/Associativity

Associative array - Wikipedia, the free encyclopedia

As a consequence, a common and important use of associative arrays is in memoization. ... The values of an associative array do not need to be the same type, although ...
http://en.wikipedia.org/wiki/Associative_array

Associative algebra: Definition from Answers.com

associative algebra ( ə′sōsē′ādiv ′aljəbrə ) ( mathematics ) An algebra in which the vector multiplication obeys the associative
http://www.answers.com/topic/associative-algebra

Associative Solutions Corporation

Associative Solutions Corporation, the associative memory model is the future of ... Privacy Statement © 2003, Associative Solutions Corporation, All Rights Reserved. ...
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associative - Definition from the Merriam-Webster Online Dictionary

Definition of associative from the Merriam-Webster Online Dictionary with audio pronunciations, thesaurus, Word of the Day, and word games.
http://www.merriam-webster.com/dictionary/associative

Hashed Associative Container

A hash function for a Hashed Associative Container X is a Unary Function whose ... A Hashed Associative Container with a small load factor is faster than one with ...
http://www.sgi.com/tech/stl/HashedAssociativeContainer.html

Sorted Associative Container

Sorted Associative Containers use an ordering relation on their keys; two keys ... The Associative Container requirements, however, merely state that the return ...
http://www.sgi.com/tech/stl/SortedAssociativeContainer.html

About ASC

Associative Solutions Corporation (ASC) provides engineering support and ... ASC engineers can help you with the premier associative CASE tool. ...
http://www.associativesolutions.com/aboutUs.php

associative property of addition and multiplication ... The associative property is a lot like the commutative property, except that it ...
http://home.europa.com/~paulg/mathmodels/associative.html

Friends of Associative Economics: Home

Friends of Associative Economics - support ongoing worldwide research with your donation. ... Associative economics refers to the shift from competitive, ...
http://www.aefriends.com/

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Q.	Commutative and associative? Which one of these properties do you think is the most practical or useful?	Related Search: Mathematics
	The commutative and associative properties. Which one of these properties do you think is the most practical or useful? You may refer to math or everyday life. If you could explain the reasons for your opinion that would be great. Support your reasons with examples if possible.
A.	An interesting question. My answer depends quite a lot on the context. If you mean in a mathematical research type of sense, then I would say associativity is more useful. We know of a good number of examples of systems which are non-commutative. For a few examples, consider matrices, quaternions, functions under composition, etc. In fact, the commutative (or "abelian") groups are somewhat boring to study in abstract algebra. On the other hand, associativity is in every example I mentioned above. Almost all of the common sets and operations have associativity. Well, I suppose subtraction and division are nonassociative, but these are inverses of addition and multiplication, so they don't really count (so far as I'm concerned). One good example of a nonassociative operation is the cross product of vectors. But that's about all that I can think of. Notice that the cross product is also noncommutative. If I think of this in a more everyday setting though, I think it's harder to say. They both play such an important role in my mental arithmetic. Really I don't think I could do most of that sort of thing without either of these properties, since they're so deeply ingrained in my mind as working for any real numbers I should happen to need to work with.

Q.	Can anyone tell me anything about selectivity in associative learning?	Related Search: Psychology
	I'm writing a psychology report on selectivity in associative learning but am struggling to find anything on selective associative learning on the internet. Does anyone know any good websites? I've tried googling it, and all that comes up is examples of how a certain drug affects it, I need something on the main principles and theory of it. Thanks.
A.	Associative Learning Associative learning takes place whenever an animal learns to associate an external event with a change in its own internal state, or a change in its behavior. In addition, an animal can also learn to associate an act that it performs with some kind of reward. The first form of learning is classical conditioning, the second form of learning is operant or instrument conditioning. A chicken with a bird feeder that comes on at night, may happen to create a shadow on the sensor that turns on the feeder in daylight. That chicken may then learn to place its head or body in a position to repeat that shadow when it is hungry and feed all the time when it is hungry. Yes, the horrible, and deadly myth of permanence in ability is killing many students and adults. We need desperately to remove this false and deadly myth of fixed intelligences and abilities and replace it with tools to continually improve and change our lives. Please Please read learning theory and Grand Hope, and also Little Girl Innocence from profile and my blogs, and also on request to all by e-mail. By showing students how their individual environments greatly affect their ability to think, learn, long-term motivation to learn, and grow mentally and emotionally, students will have much more respect and esteem for themselves and for others. By providing students with tools to approach their lives more delicately and differently to continually change and improve their lives, students will then have a continuous hope of developing in time, many if not all of the qualities they admire in others over time. Students will then have a continuous hope of changing and becoming better, newer persons with each passing day. This will reduce much hopelessness, many harmful escapes and other problems created by our horrible teachings of fixed intelligences in school such as dropouts, drug/alcohol abuse, catharsis of violence, and suicide. I would like for all students though to be more thoughtful of seeing our minds as capable of much more thinking, learning, and motivation to learn by seeing how they can approach their more delicately to greatly improve from within their ability to think and learn. Imagine a child who is able to learn more effectively. Try to see an upright rectangle representing our full ability. The young child has few if any layers of mental frictions so he has almost of all of his ability to learn. As we get older, we accumulate more layers of mental frictions. The trick we as humans need to learn is how to approach our lives more delicately to more permanently reduce layers of mental frictions to continually improve our thinking, learning, and mental/emotional health. Please read learning theory on profile, my blogs, and to all on request.

Q.	what are the differences between the commutative and associative properties?	Related Search: Homework Help
	i need to know the difference for math here is one of the questions is the property associative or commutative?
A.	associative: x+y=y+x commutative: z(x+y)=(z+x)+y

Q.	Do the commutative and associative properties hold for subtraction and division?	Related Search: Mathematics
	My teacher got pissed off at one student because she had no idea what the difference between associative and commutative now my teacher wants me and everyone else in the class to write about it. I understand this question but have no idea how to explain it in more than a sentence what sucks is I need it in 1 paragraph.
A.	You are in Luck. I am taught this for 33 years. So listen up. The commutative property of addition is a+b=b+a You can "commute" the two numbers around the + sign and still have the same answer. 4+2=2+4 The commutative property of multiplication is ab=ba meaning you can "commute" the numbers and still have the same answer. 5(6)=6(5) 30=30 There is no commutative property of subtraction or division. I will show you two counterexamples. 6-2=4 but 2-6= (-4) 4 does not equal (-4). 12 divided by 6= 2 but 6 divided by 12= 1/2 To remember this property, the word commute means to go back and forth from one place to another. Like commuting to work from home and back. It is the same distance coming and going. Also, the Commutative has C O as the first two letters. C could stand for Change and O stand for Order. Change order of the terms or numbers.6+2 the order changes to 2=6. Now, the associative property of addition and multiplication is changing the grouping symbols and leaving the numbers fixed. For example: a + (b+c)= (a+b) +c Notice the three numbers are in the same order, but the grouping is changed. 5+ ( 6+2)= (5 +6) +2 5 +8= 11+2 13+13 a(bc)=(ab)c 4x(2x6)=(4x2)x6 4x 12= 8x6 48=48 To associate with friends means to talk to them. Think of this example. Mary You Bill are standing in a row. First you talk to Mary. That would be (mary + You) + Bill Bill is still there but he is not in the conversation. Next, You turn to Bill and talk to him. That would be Mary + ( You + Bill). The three people or numbers are still there -only the grouping or the association would change. There is not associative property of subtraction or division: 10- ( 8 -2 ) does not equal ( 10 - 8 ) -2 10- ( 6)= 4 2 -2=0 4 does not equal 0. Good Luck. Try to summarize this. Do not use word for word in your explanation. Your teacher will know that you didn't write it.

Q.	How do I found out which problem is the distributive property and which is the associative property?	Related Search: Homework Help
	It's asking me which it the distributive property and associative property in these Algebra problems. Help please. 1. (a+b)+c=a+(b+c) 2. a(b+c)=ab+ac 3. a*1=a 4. a+b+c=a+c=b Ah, thank you!
A.	Associative property states that the change in grouping of three or more addends or factors does not change their sum or product Examples of Associative Property Addition (2 + 3) + 5 = 2 + (3 + 5) Whether you add 2 & 3 first or 3 & 5 first does not matter as you get the the same sum (10) both ways. Multiplication (4 . 5) . 10 = 4 . (5 . 10) Whether you multiply 4 & 5 first or 5 & 10 first does not matter as you get the the same product (200) both ways. Distributive Property states that the product of a number and a sum is equal to the sum of the individual products of the addends and the number. That is, a(b + c) = ab + ac. Examples of Distributive Property 5(3 + 1) = 5 × 3 + 5 × 1 Consider LHS: 5(3 + 1) = 5(4) = 20 Consider RHS: 5 × 3 + 5 × 1 = 15 + 5 = 20 LHS = RHS HOPE THIS HELPS!! GOOD LUCK!

Q.	Use the associative property of multiplication to write an expression that is equivalent to the following expr	Related Search: Mathematics
	Use the associative property of multiplication to write an expression that is equivalent to the following expression. 3(ab)
A.	3ab

Q.	What is the difference between commutative property and associative property?	Related Search: Mathematics
	they seem like the same thing to me!
A.	Commutative property changes the order. Example: 1+2+3=6 or 2+3+1=6 Associative property keeps the same order but changes parentheses. 1 x (2 x 3) = 6 or (1 x 2) x 3 = 6

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This article is about associativity in mathematics. For Associativity in the central processor unit memory cache, see CPU cache. For Associativity in programming languages, see operator associativity.

In mathematics, associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order that the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. Consider for instance the equation

$(5+2)+1=5+(2+1)=8 \,$

Even though the parentheses were rearranged (the left side requires adding 5 and 2 first, then adding 1 to the result, whereas the right side requires adding 2 and 1 first, then 5), the value of the expression was not altered. Since this holds true when performing addition on any real numbers, we say that "addition of real numbers is an associative operation."

Associativity is not to be confused with commutativity. Commutativity justifies changing the order or sequence of the operands within an expression while associativity does not. For example,

$(5+2)+1=5+(2+1) \,$

is an example of associativity because the parentheses were changed (and consequently the order of operations during evaluation) while the operands 5, 2, and 1 appeared in the exact same order from left to right in the expression.

$(5+2)+1=(2+5)+1 \,$

is not an example of associativity because the operand sequence changed when the 2 and 5 switched places.

Associative operations are abundant in mathematics, and in fact most algebraic structures explicitly require their binary operations to be associative. However, many important and interesting operations are non-associative; one common example would be the vector cross product.

1 Definition
2 Examples
3 Non-associativity
4 More examples
5 See also

[edit] Definition

Formally, a binary operation $*\!\!\!$ on a set S is called associative if it satisfies the associative law:

$(x * y) * z=x * (y * z)\qquad\mbox{for all }x,y,z\in S.$

Using * to denote a binary operation performed on a set

$(xy)z=x(yz) = xyz \qquad\mbox{for all }x,y,z\in S.$

An example of multiplicative associativity

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of $*\!\!\!$ operations. Thus, when $*\!\!\!$ is associative, the evaluation order can therefore be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:

x y z,

However, it is important to remember that changing the order of operations does not involve or permit changing the actual operations themselves by moving the operands around within the expression.

[edit] Examples

Some examples of associative operations include the following.

In arithmetic, addition and multiplication of real numbers are associative; i.e.,

$\left. \begin{matrix} (x+y)+z=x+(y+z)=x+y+z\quad \\ (x\,y)z=x(y\,z)=x\,y\,z\qquad\qquad\qquad\quad\ \ \, \end{matrix} \right\} \mbox{for all }x,y,z\in\mathbb{R}.$

Addition and multiplication of complex numbers and quaternions is associative. Addition of octonions is also associative, but multiplication of octonions is non-associative.

The greatest common divisor and least common multiple functions act associatively.

$\left. \begin{matrix} \operatorname{gcd}(\operatorname{gcd}(x,y),z)= \operatorname{gcd}(x,\operatorname{gcd}(y,z))= \operatorname{gcd}(x,y,z)\ \quad \\ \operatorname{lcm}(\operatorname{lcm}(x,y),z)= \operatorname{lcm}(x,\operatorname{lcm}(y,z))= \operatorname{lcm}(x,y,z)\quad \end{matrix} \right\}\mbox{ for all }x,y,z\in\mathbb{Z}.$

Because linear transformations are functions that can be represented by matrices with matrix multiplication being the representation of functional composition, one can immediately conclude that matrix multiplication is associative.

Taking the intersection or the union of sets:

$\left. \begin{matrix} (A\cap B)\cap C=A\cap(B\cap C)=A\cap B\cap C\quad \\ (A\cup B)\cup C=A\cup(B\cup C)=A\cup B\cup C\quad \end{matrix} \right\}\mbox{for all sets }A,B,C.$

If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

$(f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h\qquad\mbox{for all }f,g,h\in S.$

Slightly more generally, given four sets M, N, P and Q, with h: M to N, g: N to P, and f: P to Q, then

$(f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h$

as before. In short, composition of maps is always associative.

Consider a set with three elements, A, B, and C. The following operation:

×	A	B	C
+
A	A	A	A
B	A	B	C
C	A	A	A

is associative. Thus, for example, A(BC)=(AB)C. This mapping is not commutative.

[edit] Non-associativity

A binary operation $*$ on a set S that does not satisfy the associative law is called non-associative. Symbolically,

$(x*y)*z\ne x*(y*z)\qquad\mbox{for some }x,y,z\in S.$

For such an operation the order of evaluation does matter. Subtraction, division and exponentiation are well-known examples of non-associative operations:

$\begin{matrix} (5-3)-2\ne 5-(3-2)\quad \\ (4/2)/2\ne 4/(2/2)\qquad\qquad \\ 2^{(1^2)}\ne (2^1)^2.\quad\qquad\qquad \end{matrix}$

In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears more than once in an expression. However, mathematicians agree on a particular order of evaluation for several common non-associative operations. This is simply a syntactical convention to avoid parentheses.

A left-associative operation is a non-associative operation that is conventionally evaluated from left to right, i.e.,

$\left. \begin{matrix} x*y*z=(x*y)*z\qquad\qquad\quad\, \\ w*x*y*z=((w*x)*y)*z\quad \\ \mbox{etc.}\qquad\qquad\qquad\qquad\qquad\qquad\ \ \, \end{matrix} \right\} \mbox{for all }w,x,y,z\in S$

while a right-associative operation is conventionally evaluated from right to left:

$\left. \begin{matrix} x*y*z=x*(y*z)\qquad\qquad\quad\, \\ w*x*y*z=w*(x*(y*z))\quad \\ \mbox{etc.}\qquad\qquad\qquad\qquad\qquad\qquad\ \ \, \end{matrix} \right\} \mbox{for all }w,x,y,z\in S$

Both left-associative and right-associative operations occur; examples are given below.

[edit] More examples

Left-associative operations include the following.

Subtraction and division of real numbers:

$x-y-z=(x-y)-z\qquad\mbox{for all }x,y,z\in\mathbb{R};$

$x/y/z=(x/y)/z\qquad\qquad\quad\mbox{for all }x,y,z\in\mathbb{R}\mbox{ with }y\ne0,z\ne0.$

Right-associative operations include the following.

Exponentiation of real numbers:

$x^{y^z}=x^{(y^z)}.\,$

The reason exponentiation is right-associative is that a repeated left-associative exponentiation operation would be less useful. Multiple appearances could (and would) be rewritten with multiplication:

$(x^y)^z=x^{(yz)}.\,$

Non-associative operations for which no conventional evaluation order is defined include the following.

Taking the pairwise average of real numbers:

${(x+y)/2+z\over2}\ne{x+(y+z)/2\over2}\ne{x+y+z\over3}\qquad\mbox{for some }x,y,z\in\mathbb{R}.$

Taking the relative complement of sets:

$(A\backslash B)\backslash C\ne A\backslash (B\backslash C)\qquad\mbox{for some sets }A,B,C.$

$Venn diagram of the relative complements (A\B)\C and A\(B\C)$

The green part in the left Venn diagram represents $(A\backslash B)\backslash C$ . The green part in the right Venn diagram represents $A\backslash(B\backslash C)$ .

Using right-associative notation for material conditional can be motivated e.g. by Curry-Howard correspondence: see e.g. comparison of the first two axioms of the Hilbert-style deduction system with basic combinators of combinatory logic.

[edit] See also

Look up Associativity in
Wiktionary, the free dictionary.

A semigroup is a set with a closed associative binary operation.
Commutativity and distributivity are two other frequently discussed properties of binary operations.
Power associativity and alternativity are weak forms of associativity.

Retrieved from "http://en.wikipedia.org/wiki/Associativity"

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