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Encyclopedia > 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 of operations 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 Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, an operand is one of the inputs (arguments) of an operator. ... For the round brackets used in punctuation, often called parentheses, see bracket. ...

(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 as numbers that can be given by an infinite decimal representation, such as 2. ...


Associativity is not to be confused with the commutativity. Commutativity justifies changing the order or sequence of the operands within an expression while associativity does not. For example, A map or binary operation from a set to a set is said to be commutative if, (A common example in school-math is the + function: , thus the + function is commutative) Otherwise, the operation is noncommutative. ...

(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. In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... In mathematics, the cross product is a binary operation on vectors in three dimensions. ...

Contents

Definition

Formally, a binary operation *!!! on a set S is called associative if it satisfies the associative law: In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

(x*y)*z=x*(y*z)qquadmbox{for all }x,y,zin 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.

 left. begin{matrix} (x+y)+z=x+(y+z)=x+y+zquad  (x,y)z=x(y,z)=x,y,zqquadqquadqquadquad  , end{matrix} right} mbox{for all }x,y,zinmathbb{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,zinmathbb{Z}.
 left. begin{matrix} (Acap B)cap C=Acap(Bcap C)=Acap Bcap Cquad  (Acup B)cup C=Acup(Bcup C)=Acup Bcup Cquad 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:
(fcirc g)circ h=fcirc(gcirc h)=fcirc gcirc hqquadmbox{for all }f,g,hin 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
(fcirc g)circ h=fcirc(gcirc h)=fcirc gcirc 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. Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... In arithmetic and number theory, the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. ... This article gives an overview of the various ways to perform matrix multiplication. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... 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, all elements of B that also belong to A), but no other elements. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...


Non-associativity

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

(x*y)*zne x*(y*z)qquadmbox{for some }x,y,zin S.

For such an operation the order of evaluation does matter. Subtraction, division and exponentiation are well-known examples of non-associative operations: 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... “Exponent” redirects here. ...

 begin{matrix} (5-3)-2ne 5-(3-2)quad  (4/2)/2ne 4/(2/2)qquadqquad  2^{(1^2)}ne (2^1)^2.quadqquadqquad 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. In arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ...


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)*zqquadqquadquad,  w*x*y*z=((w*x)*y)*zquad  mbox{etc.}qquadqquadqquadqquadqquadqquad  , end{matrix} right} mbox{for all }w,x,y,zin S

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

 left. begin{matrix} x*y*z=x*(y*z)qquadqquadquad,  w*x*y*z=w*(x*(y*z))quad  mbox{etc.}qquadqquadqquadqquadqquadqquad  , end{matrix} right} mbox{for all }w,x,y,zin 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)-zqquadmbox{for all }x,y,zinmathbb{R};
x/y/z=(x/y)/zqquadqquadquadmbox{for all }x,y,zinmathbb{R}mbox{ with }yne0,zne0.

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)}.,

Non-associative operations for which no conventional evaluation order is defined include the following. “Exponent” redirects here. ...

  • Taking the pairwise average of real numbers:
{(x+y)/2+zover2}ne{x+(y+z)/2over2}ne{x+y+zover3}qquadmbox{for some }x,y,zinmathbb{R}.
(Abackslash B)backslash Cne Abackslash (Bbackslash C)qquadmbox{for some sets }A,B,C.

The green part in the left Venn diagram represents (AB)C. The green part in the right Venn diagram represents A(BC)
In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... Venn diagram of relative complements (AB)C and A(BC). ... A Venn diagram of sets A, B, and C Venn diagrams are illustrations used in the branch of mathematics known as set theory. ...

  • 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.

The material conditional, also known as the truth functional conditional, expresses a property of certain conditionals in logic. ... The Curry-Howard correspondence is the close relationship between computer programs and mathematical proofs; the correspondence is also known as the Curry-Howard isomorphism, or the formulae-as-types correspondence. ... The phrase Hilbert-style deduction system denotes a specific formalization of notion deduction in first-order logic [1], attributed to Gottlob Frege and David Hilbert. ...

See also

Look up Associativity in
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  Results from FactBites:
 
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Examples of associative operations are addition and multiplication over the integers (or reals), or addition or multiplication over
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