TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the associative property is a property of some
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, which means that rearranging the parentheses in an expression will not change the result. In
propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...
, associativity is a valid
rule of replacement In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
for
expressions Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphor#Common types, Metaphorical expression, a parti ...
in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the
operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical or dental specialty that ...
are performed does not matter as long as the sequence of the
operand In mathematics an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Example The following arithmetic expression shows an example of operators and operands: :3 + 6 = 9 In the above example ...
s is not changed. That is, (after rewriting the expression with parentheses and in infix notation if necessary) rearranging the
parentheses A bracket is either of two tall fore- or back-facing punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding ...
in such an expression will not change its value. Consider the following equations: :$\left(2 + 3\right) + 4 = 2 + \left(3 + 4\right) = 9 \,$ :$2 \times \left(3 \times 4\right) = \left(2 \times 3\right) \times 4 = 24 .$ Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, it can be said that "addition and multiplication of real numbers are associative operations". Associativity is not the same as
commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, which addresses whether the order of two
operand In mathematics an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Example The following arithmetic expression shows an example of operators and operands: :3 + 6 = 9 In the above example ...
s affects the result. For example, the order does not matter in the multiplication of real numbers, that is, , so we say that the multiplication of real numbers is a commutative operation. However, operations such as
function composition In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
and
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

are associative, but (generally) not commutative. Associative operations are abundant in mathematics; in fact, many
algebraic structure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s (such as
semigroups In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation. The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...
and
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...
) explicitly require their binary operations to be associative. However, many important and interesting operations are non-associative; some examples include
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

,
exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
, and the
vector cross product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
. In contrast to the theoretical properties of real numbers, the addition of
floating point In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
numbers in computer science is not associative, and the choice of how to associate an expression can have a significant effect on rounding error.

# Definition

Formally, a
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
∗ on a set ''S'' is called associative if it satisfies the associative law: :(''x'' ∗ ''y'') ∗ ''z'' = ''x'' ∗ (''y'' ∗ ''z'') for all ''x'', ''y'', ''z'' in ''S''. Here, ∗ is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbol (
juxtaposition Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc. Speech Juxtaposition in literary terms is the showing ...
) as for
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

. :(''xy'')''z'' = ''x''(''yz'') = ''xyz'' for all ''x'', ''y'', ''z'' in ''S''. The associative law can also be expressed in functional notation thus: .

# Generalized associative law

If a binary operation is associative, repeated application of the operation produces the same result regardless of how valid pairs of parentheses are inserted in the expression. This is called the generalized associative law. For instance, a product of four elements may be written, without changing the order of the factors, in five possible ways: : $\left(\left(ab\right)c\right)d$ : $\left(ab\right)\left(cd\right)$ : $\left(a\left(bc\right)\right)d$ : $a\left(\left(bc\right)d\right)$ : $a\left(b\left(cd\right)\right)$ If the product operation is associative, the generalized associative law says that all these formulas will yield the same result. So unless the formula with omitted parentheses already has a different meaning (see below), the parentheses can be considered unnecessary and "the" product can be written unambiguously as :$abcd.$ As the number of elements increases, the number of possible ways to insert parentheses grows quickly, but they remain unnecessary for disambiguation. An example where this does not work is the
logical biconditional In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stat ...
$\leftrightarrow$. It is associative, thus A$\leftrightarrow$(B$\leftrightarrow$C) is equivalent to (A$\leftrightarrow$B)$\leftrightarrow$C, but A$\leftrightarrow$B$\leftrightarrow$C most commonly means (A$\leftrightarrow$B and B$\leftrightarrow$C), which is not equivalent.

# Examples

Some examples of associative operations include the following. * The
concatenation In formal language theory In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt ...
of the three strings "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. The two methods produce the same result; string concatenation is associative (but not commutative). * In
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
,
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s are associative; i.e., :: $\left. \begin \left(x+y\right)+z=x+\left(y+z\right)=x+y+z\quad \\ \left(x\,y\right)z=x\left(y\,z\right)=x\,y\,z\qquad\qquad\qquad\quad\ \ \, \end \right\\right\} \mboxx,y,z\in\mathbb.$ :Because of associativity, the grouping parentheses can be omitted without ambiguity. * The trivial operation (that is, the result is the first argument, no matter what the second argument is) is associative but not commutative. Likewise, the trivial operation (that is, the result is the second argument, no matter what the first argument is) is associative but not commutative. * Addition and multiplication of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s and
quaternion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

octonion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s is also associative, but multiplication of octonions is non-associative. * The
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

and
least common multiple In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ...

functions act associatively. :: $\left. \begin \operatorname\left(\operatorname\left(x,y\right),z\right)= \operatorname\left(x,\operatorname\left(y,z\right)\right)= \operatorname\left(x,y,z\right)\ \quad \\ \operatorname\left(\operatorname\left(x,y\right),z\right)= \operatorname\left(x,\operatorname\left(y,z\right)\right)= \operatorname\left(x,y,z\right)\quad \end \right\\right\}\mboxx,y,z\in\mathbb.$ * Taking the
intersection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
or the union of sets: :: $\left. \begin \left(A\cap B\right)\cap C=A\cap\left(B\cap C\right)=A\cap B\cap C\quad \\ \left(A\cup B\right)\cup C=A\cup\left(B\cup C\right)=A\cup B\cup C\quad \end \right\\right\}\mboxA,B,C.$ * If ''M'' is some set and ''S'' denotes the set of all functions from ''M'' to ''M'', then the operation of
function composition In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
on ''S'' is associative: :: $\left(f\circ g\right)\circ h=f\circ\left(g\circ h\right)=f\circ g\circ h\qquad\mboxf,g,h\in S.$ * 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 :: $\left(f\circ g\right)\circ h=f\circ\left(g\circ h\right)=f\circ g\circ h$ : as before. In short, composition of maps is always associative. * Consider a set with three elements, A, B, and C. The following operation: : :is associative. Thus, for example, A(BC)=(AB)C = A. This operation is not commutative. * Because
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object. Fo ...
represent
linear function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s, and
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

represents function composition, one can immediately conclude that matrix multiplication is associative.

# Propositional logic

## Rule of replacement

In standard truth-functional propositional logic, ''association'', or ''associativity'' are two valid rules of replacement. The rules allow one to move parentheses in
logical expressions Logic (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 milli ...
in logical proofs. The rules (using logical connectives notation) are: :$\left(P \lor \left(Q \lor R\right)\right) \Leftrightarrow \left(\left(P \lor Q\right) \lor R\right)$ and :$\left(P \land \left(Q \land R\right)\right) \Leftrightarrow \left(\left(P \land Q\right) \land R\right),$ where "$\Leftrightarrow$" is a
metalogic Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how formal system, logical systems can be used to construct Validity (logic), valid and soundness, sound arguments, metalogic studies the properties of logical systems.Har ...
al
symbol A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), m ...
representing "can be replaced in a
proof Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
with."

## Truth functional connectives

''Associativity'' is a property of some
logical connective In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents sta ...
s of truth-functional
propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...
. The following
logical equivalence In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statemen ...
s demonstrate that associativity is a property of particular connectives. The following are truth-functional tautologies. Associativity of disjunction: :$\left(\left(P \lor Q\right) \lor R\right) \leftrightarrow \left(P \lor \left(Q \lor R\right)\right)$ :$\left(P \lor \left(Q \lor R\right)\right) \leftrightarrow \left(\left(P \lor Q\right) \lor R\right)$ Associativity of conjunction: :$\left(\left(P \land Q\right) \land R\right) \leftrightarrow \left(P \land \left(Q \land R\right)\right)$ :$\left(P \land \left(Q \land R\right)\right) \leftrightarrow \left(\left(P \land Q\right) \land R\right)$ Associativity of equivalence: :$\left(\left(P \leftrightarrow Q\right) \leftrightarrow R\right) \leftrightarrow \left(P \leftrightarrow \left(Q \leftrightarrow R\right)\right)$ :$\left(P \leftrightarrow \left(Q \leftrightarrow R\right)\right) \leftrightarrow \left(\left(P \leftrightarrow Q\right) \leftrightarrow R\right)$ Joint denial is an example of a truth functional connective that is ''not'' associative.

# Non-associative operation

A binary operation $*$ on a set ''S'' that does not satisfy the associative law is called non-associative. Symbolically, :$\left(x*y\right)*z\ne x*\left(y*z\right)\qquad\mboxx,y,z\in S.$ For such an operation the order of evaluation ''does'' matter. For example: *
Subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

:$\left(5-3\right)-2 \, \ne \, 5-\left(3-2\right)$ *
Division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
:$\left(4/2\right)/2 \, \ne \, 4/\left(2/2\right)$ *
Exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
:$2^ \, \ne \, \left(2^1\right)^2$ Also note that infinite sums are not generally associative, for example: :$\left(1+-1\right)+\left(1+-1\right)+\left(1+-1\right)+\left(1+-1\right)+\left(1+-1\right)+\left(1+-1\right)+\dots \, = \, 0$ whereas :$1+\left(-1+1\right)+\left(-1+1\right)+\left(-1+1\right)+\left(-1+1\right)+\left(-1+1\right)+\left(-1+1\right)+\dots \, = \, 1$ The study of non-associative structures arises from reasons somewhat different from the mainstream of classical algebra. One area within
non-associative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...
that has grown very large is that of
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s. There the associative law is replaced by the
Jacobi identity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. Lie algebras abstract the essential nature of
infinitesimal transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, and have become ubiquitous in mathematics. There are other specific types of non-associative structures that have been studied in depth; these tend to come from some specific applications or areas such as
combinatorial mathematics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other are ...
. Other examples are
quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group (mathematics), group in the sense that "division (mathematics), division" is always possible. Quasigroups differ from groups mainly in that th ...
,
quasifield In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, non-associative ring,
non-associative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...
and commutative non-associative magmas.

## Nonassociativity of floating point calculation

In mathematics, addition and multiplication of real numbers is associative. By contrast, in computer science, the addition and multiplication of
floating point In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
numbers is ''not'' associative, as rounding errors are introduced when dissimilar-sized values are joined together. To illustrate this, consider a floating point representation with a 4-bit
mantissa Mantissa () may refer to: * Mantissa (logarithm), the fractional part of the common (base-10) logarithm * Mantissa (floating point number) The significand (also mantissa or coefficient, sometimes also argument, or ambiguously fraction or characte ...
:
(1.0002×20 + 1.0002×20) + 1.0002×24 = 1.0002×2 + 1.0002×24 = 1.002×24
1.0002×20 + (1.0002×20 + 1.0002×24) = 1.0002×2 + 1.0002×24 = 1.002×24 Even though most computers compute with a 24 or 53 bits of mantissa, this is an important source of rounding error, and approaches such as the
Kahan summation algorithm In numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite-decimal precision, precision floating-point numbers, compared t ...
are ways to minimise the errors. It can be especially problematic in parallel computing.

)

## Notation for non-associative operations

In general, parentheses must be used to indicate the order of operations, order of evaluation if a non-associative operation appears more than once in an expression (unless the notation specifies the order in another way, like $\dfrac$). However,
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

s 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., :$\left. \begin x*y*z=\left(x*y\right)*z\qquad\qquad\quad\, \\ w*x*y*z=\left(\left(w*x\right)*y\right)*z\quad \\ \mbox\qquad\qquad\qquad\qquad\qquad\qquad\ \ \, \end \right\\right\} \mboxw,x,y,z\in S$ while a right-associative operation is conventionally evaluated from right to left: :$\left. \begin x*y*z=x*\left(y*z\right)\qquad\qquad\quad\, \\ w*x*y*z=w*\left(x*\left(y*z\right)\right)\quad \\ \mbox\qquad\qquad\qquad\qquad\qquad\qquad\ \ \, \end \right\\right\} \mboxw,x,y,z\in S$ Both left-associative and right-associative operations occur. Left-associative operations include the following: * Subtraction and division of real numbers:Virginia Department of Education
Using Order of Operations and Exploring Properties
section 9
Bronstein: :de:Taschenbuch der Mathematik, pages 115-120, chapter: 2.4.1.1, ::$x-y-z=\left(x-y\right)-z$ ::$x/y/z=\left(x/y\right)/z$ * Function application: ::$\left(f \, x \, y\right) = \left(\left(f \, x\right) \, y\right)$ :This notation can be motivated by the
currying In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
isomorphism. Right-associative operations include the following: *
Exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
of real numbers in superscript notation: ::$x^=x^$ :Exponentiation is commonly used with brackets or right-associatively because a repeated left-associative exponentiation operation is of little use. Repeated powers would mostly be rewritten with multiplication: ::$\left(x^y\right)^z=x^$ :Formatted correctly, the superscript inherently behaves as a set of parentheses; e.g. in the expression $2^$ the addition is performed
before Before is the opposite of after, and may refer to: * Before (album), ''Before'' (album) by Gold Panda * Before (song), "Before" (song) by the Pet Shop Boys * Before (short story), "Before" (short story) by Gael Baudino * The Before film trilogy by ...

the exponentiation despite there being no explicit parentheses $2^$ wrapped around it. Thus given an expression such as $x^$, the full exponent $y^z$ of the base $x$ is evaluated first. However, in some contexts, especially in handwriting, the difference between $^z=\left(x^y\right)^z$, $x^=x^$ and $x^=x^$ can be hard to see. In such a case, right-associativity is usually implied. *
Function definition Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriented ...
::$\mathbb \rarr \mathbb \rarr \mathbb = \mathbb \rarr \left(\mathbb \rarr \mathbb\right)$ ::$x \mapsto y \mapsto x - y = x \mapsto \left(y \mapsto x - y\right)$ :Using right-associative notation for these operations can be motivated by the
Curry–Howard correspondence In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relati ...
and by the
currying In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
isomorphism. Non-associative operations for which no conventional evaluation order is defined include the following. * Exponentiation of real numbers in infix notation:Exponentiation Associativity and Standard Math Notation
Codeplea. 23 August 2016. Retrieved 20 September 2016.
::$\left(x^\wedge y\right)^\wedge z\ne x^\wedge\left(y^\wedge z\right)$ * Knuth's up-arrow operators: ::$a \uparrow \uparrow \left(b \uparrow \uparrow c\right) \ne \left(a \uparrow \uparrow b\right) \uparrow \uparrow c$ ::$a \uparrow \uparrow \uparrow \left(b \uparrow \uparrow \uparrow c\right) \ne \left(a \uparrow \uparrow \uparrow b\right) \uparrow \uparrow \uparrow c$ * Taking the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

of three vectors: ::$\vec a \times \left(\vec b \times \vec c\right) \neq \left(\vec a \times \vec b \right) \times \vec c \qquad \mbox \vec a,\vec b,\vec c \in \mathbb^3$ * Taking the pairwise
average In colloquial language, an average is a single number taken as representative of a non-empty list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divide ...

of real numbers: ::$\ne \qquad \mboxx,y,z\in\mathbb \mboxx\ne z.$ * Taking the
relative complement In set theory, the complement of a Set (mathematics), set , often denoted by (or ), are the Element (mathematics), elements not in . When all sets under consideration are considered to be subsets of a given set , the absolute complement of is t ...
of sets $\left(A\backslash B\right)\backslash C$ is not the same as $A\backslash \left(B\backslash C\right)$. (Compare
material nonimplication Material nonimplication or abjunction (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Thro ...
in logic.)

* Light's associativity test *
Telescoping series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, the use of addition associativity for cancelling terms in an infinite
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
* A
semigroup In mathematics, a semigroup is an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
is a set with an associative binary operation. *
Commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

and
distributivity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
are two other frequently discussed properties of binary operations. *
Power associativityIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
,
alternativity In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
,
flexibility Stiffness is the extent to which an object resists deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (mechanics), such changes co ...
and N-ary associativity are weak forms of associativity. *
Moufang identities In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
also provide a weak form of associativity.

# References

{{reflist Properties of binary operations Elementary algebra Functional analysis Rules of inference