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associative law

 

Two closely related laws of number operations. In symbols, they are stated: a + (b + c) = (a + b) + c, and a(bc) = (ab)c. Stated in words: The terms or factors may be associated in any way desired and the result will be the same. This holds for the numbers generally encountered: positive and negative, integral and fractional, rational and irrational, real and imaginary. Exceptions occur (e.g., in nonassociative algebras and divergent infinite series). See also commutative law, distributive law.

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Columbia Encyclopedia: associative law
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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). Multiplication of numbers is also associative, i.e., (a×bc=a×(b×c). In general, any binary operation, symbolized by ◦, joining mathematical entities A, B, and C obeys the associative law if (AB)◦C=A◦(BC) for all possible choices of A, B, and C. Not all operations are associative. For example, ordinary division is not, since (60÷12)÷3=5÷3=5/3, while 60÷(12÷3)=60÷4=15. 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.


Wikipedia: Associativity
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In mathematics, associativity is a property that a binary operation can have. It means that, within an expression containing two or more occurrences in a row of the same associative operator, 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; in fact, many algebraic structures (such as semigroups and categories) 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.

Contents

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:

xyz,

However, it is important to remember that changing the order of operations does not involve or permit moving the operands around within the expression; the sequence of operands is always unchanged.

A very different perspective is obtained by rephrasing associativity using functional notation: f(f(x,y),z) = f(x,f(y,z)): when expressed in this form, associativity becomes less obvious.

Associativity can be generalized to n-ary operations. Ternary associativity is (abc)de = a(bcd)e = ab(cde), i.e. the string abcde with any three adjacent elements bracketed. N-ary associativity is a string of length n+(n-1) with any n adjacent elements bracketed[1].

Examples

Some examples of associative operations include the following.

  • The prototypical example of an associative operation is string concatenation: the concatenation of "hello", ", ", "world" can be computed by concatenating the first two strings (giving "hello, ") and appending the third string ("world"), or by joining the second and third string (giving ", world") and concatenating the first string ("hello") with the result.

\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.


\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.


\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.

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}

The study of non-associative structures arises from reasons somewhat different from the mainstream of classical algebra. One area within non-associative algebra that has grown very large is that of Lie algebras. There the associative law is replaced by the Jacobi identity. Lie algebras abstract the essential nature of infinitesimal transformations, and have become ubiquitous in mathematics. They are an example of non-associative algebras.

There are other specific types of non-associative structures that have been studied in depth. They tend to come from some specific applications. Some of these arise in combinatorial mathematics. Other examples: Quasigroup, Quasifield, Nonassociative ring.

Notation for non-associative operations

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 notational 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. 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.
  • Function application:
(f \, x \, y) = ((f \, x) \, y)
This notation can be motivated by the currying isomorphism.

Right-associative operations include the following:

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)}.\,
\mathbb{Z} \rarr \mathbb{Z} \rarr \mathbb{Z} = \mathbb{Z} \rarr (\mathbb{Z} \rarr \mathbb{Z})
x \mapsto y \mapsto x - y = x \mapsto (y \mapsto x - y)
Using right-associative notation for these operations can be motivated by the Curry-Howard correspondence and by the currying isomorphism.

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} \qquad \mbox{for some }x,y,z\in\mathbb{R}.
(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).

See also

References

  1. ^ Dudek, W.A. (2001), "On some old problems in n-ary groups", Quasigroups and Related Systems 8: 15–36, http://www.quasigroups.eu/contents/contents8.php?m=trzeci .

 
 

 

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