Associativity

This article is about associativity in mathematics. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and For associativity in central processor unit memory cache architecture see CPU cache. For associativity in programming languages see operator associativity. In Programming languages, the associativity of an Operator is a property that determines how operators of the same precedence are grouped in the absence

In mathematics, associativity is a property that a binary operation can have. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two It means that, within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed. In Mathematics, an operand is one of the inputs (arguments of an Operator. 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 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. In Mathematics, the real numbers may be described informally in several different ways "

Associativity is not to be confused with commutativity. In Mathematics, commutativity is the ability to change the order of something without changing the end result 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. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, However, many important and interesting operations are non-associative; one common example would be the vector cross product. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

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

Definition

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

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

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.

Examples

Some examples of associative operations include the following.

In arithmetic, addition and multiplication of real numbers are associative; i. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Addition is the mathematical process of putting things together In Mathematics, the real numbers may be described informally in several different ways 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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Addition of octonions is also associative, but multiplication of octonions is non-associative. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real

The greatest common divisor and least common multiple functions act associatively. In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero In Arithmetic and Number theory, the least common multiple or lowest common multiple ( lcm) or smallest common multiple of two

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

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. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix

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 Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, a composite function represents the application of one function to the results of another 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:

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

In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears more than once in an expression. Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Algebra and Computer programming, when a number or expression is both preceded and followed by a Binary operation, a rule is required for which operation However, mathematicians agree on a particular order of evaluation for several common non-associative operations. A mathematician is a person whose primary area of study and research is the field of Mathematics. 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.

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\B)\C. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation Venn diagrams or set diagrams are Diagrams that show all hypothetically possible Logical relations between a finite collection of sets (groups The green part in the right Venn diagram represents A\(B\C).

Using right-associative notation for material conditional can be motivated e. The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain Conditionals in Logic g. by Curry-Howard correspondence: see e. The Curry-Howard correspondence is the direct relationship between computer programs and mathematical proofs g. comparison of the first two axioms of the Hilbert-style deduction system with basic combinators of combinatory logic. In Logic, especially Mathematical logic, a Hilbert-style deduction system is a type of system of formal deduction attributed

Dictionary

-noun

(algebra) the condition of being associative

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