View source for Associative property

{{Short description|Property of a mathematical operation}}
{{Redirect-multi|3|Associativity|Associative|non-associative|other uses|Associativity (disambiguation)|associative and non-associative learning|Learning#Types}}
{{More citations needed|date=June 2009}}
{{Infobox mathematical statement
| name = Associative property
| image = [[File:Associativity of binary operations (without question marks).svg|300px|class=skin-invert-image]]
| caption = A visual graph representing associative operations; <math>(x\circ y)\circ z = x\circ(y\circ z)</math>
| type = [[Principle|Law]], [[rule of replacement]]
| field = {{Plainlist|
* [[Elementary algebra]]
* [[Boolean algebra]]
* [[Set theory]]
* [[Linear algebra]]
* [[Propositional calculus]]
}}
| statement =
| symbolic statement = {{Plainlist|
# Elementary algebra
#: <math>(x \,*\, y) \,*\, z = x \,*\, (y \,*\, z) \forall x,y,z \in S</math>
# Propositional calculus
#:<math>(P \lor (Q \lor R)) \Leftrightarrow ((P \lor Q) \lor R)</math>
#:<math>(P \land (Q \land R)) \Leftrightarrow ((P \land Q) \land R),</math>
}}
}}

In [[mathematics]], the '''associative property'''<ref>
{{cite book
|last=Hungerford
|first=Thomas W.
|year=1974 |edition=1st
|title=Algebra
|page=24
|publisher=[[Springer Science+Business Media|Springer]]
|isbn=978-0387905181
|quote=Definition 1.1 (i) a(bc) = (ab)c for all a, b, c in G.}}</ref> is a property of some [[binary operation]]s that rearranging the [[parentheses]] in an expression will not change the result. In [[propositional logic]], '''associativity''' is a [[validity (logic)|valid]] [[rule of replacement]] for [[well-formed formula|expressions]] in [[Formal proof|logical proofs]].

Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the [[Operation (mathematics)|operations]] are performed does not matter as long as the sequence of the [[operand]]s is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations:

<math display="block">\begin{align}
(2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\
2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 .
\end{align}</math>

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]]s, it can be said that "addition and multiplication of real numbers are associative operations".

Associativity is not the same as [[commutativity]], which addresses whether the order of two operands affects the result. For example, the order does not matter in the multiplication of real numbers, that is, {{math|1={{var|a}} × {{var|b}} = {{var|b}} × {{var|a}}}}, so we say that the multiplication of real numbers is a commutative operation. However, operations such as [[function composition]] and [[matrix multiplication]] are associative, but not (generally) commutative.

Associative operations are abundant in mathematics; in fact, many [[algebraic structure]]s (such as [[semigroup (mathematics)|semigroups]] and [[category (mathematics)|categories]]) explicitly require their binary operations to be associative.  However, many important and interesting operations are non-associative; some examples include [[subtraction]], [[exponentiation]], and the [[vector cross product]]. In contrast to the theoretical properties of real numbers, the addition of [[floating point]] 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 ==
[[File:Semigroup_associative.svg|thumbnail|A binary operation ∗ on the set ''S'' is associative when [[Commutative diagram|this diagram commutes]]. That is, when the two paths from {{math|{{var|S}}×{{var|S}}×{{var|S}}}} to {{mvar|S}} [[Function composition|compose]] to the same function from {{math|{{var|S}}×{{var|S}}×{{var|S}}}} to {{mvar|S}}.]]
Formally, a [[binary operation]] <math>\ast</math> on a [[Set (mathematics)|set]] {{mvar|S}} is called '''associative''' if it satisfies the '''associative law''':
:<math>(x \ast y) \ast z = x \ast (y \ast z)</math>, for all <math>x,y,z</math> in {{mvar|S}}.

Here, ∗ is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbol ([[Juxtaposition#Mathematics|juxtaposition]]) as for [[multiplication]].
:<math>(xy)z = x(yz)</math>, for all <math>x,y,z</math> in {{mvar|S}}.

The associative law can also be expressed in functional notation thus: <math>(f \circ (g \circ h))(x) = ((f \circ g) \circ h)(x)</math>

== Generalized associative law ==
[[Image:Tamari lattice.svg|thumb|In the absence of the associative property, five factors {{mvar|a}}, {{mvar|b}},{{mvar|c}}, {{mvar|d}}, {{mvar|e}} result in a [[Tamari lattice]] of order four, possibly different products.]]
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.<ref>{{cite book |last=Durbin |first=John R. |title=Modern Algebra: an Introduction |year=1992 |publisher=Wiley |location=New York |isbn=978-0-471-51001-7 |page=78 |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000258.html |edition=3rd |quote=If <math>a_1, a_2, \dots, a_n \,\, (n \ge 2)</math> are elements of a set with an associative operation, then the product <math>a_1 a_2 \cdots a_n</math> is unambiguous; this is, the same element will be obtained regardless of how parentheses are inserted in the product.}}</ref> This is called the '''generalized associative law'''.

The number of possible bracketings is just the [[Catalan number]], <math>C_n</math>
, for ''n'' operations on ''n+1'' values. For instance, a product of 3 operations on 4 elements may be written (ignoring permutations of the arguments), in <math>C_3 = 5</math> possible ways:

*<math>((ab)c)d</math>
*<math>(a(bc))d</math>
*<math>a((bc)d)</math>
*<math>a(b(cd))</math>
*<math>(ab)(cd)</math>

If the product operation is associative, the generalized associative law says that all these expressions will yield the same result. So unless the expression with omitted parentheses already has a different meaning (see below), the parentheses can be considered unnecessary and "the" product can be written unambiguously as
:<math>abcd</math>
As the number of elements increases, the [[Catalan number#Applications in combinatorics|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]] {{math|↔}}. It is associative; thus, {{math|{{var|A}} ↔ ({{var|B}} ↔ {{var|C}})}} is equivalent to {{math|({{var|A}} ↔ {{var|B}}) ↔ {{var|C}}}}, but {{math|{{var|A}} ↔ {{var|B}} ↔ {{var|C}}}} most commonly means {{math|({{var|A}} ↔ {{var|B}}) and ({{var|B}} ↔ {{var|C}})}}, which is not equivalent.

== Examples ==
[[File:Associativity of real number addition.svg|thumb|The addition of real numbers is associative.]]
Some examples of associative operations include the following.

{{unordered list
|1= The [[string concatenation|concatenation]] of the three strings <code>"hello"</code>, <code>" "</code>, <code>"world"</code> can be computed by concatenating the first two strings (giving <code>"hello "</code>) and appending the third string (<code>"world"</code>), or by joining the second and third string (giving <code>" world"</code>) and concatenating the first string (<code>"hello"</code>) with the result. The two methods produce the same result; string concatenation is associative (but not commutative).
|2= In [[arithmetic]], [[addition]] and [[multiplication]] of [[real number]]s are associative; i.e.,

<math display="block">
\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}.
</math>

Because of associativity, the grouping parentheses can be omitted without ambiguity.
|3= The trivial operation {{math|1={{var|x}} ∗ {{var|y}} = {{var|x}}}} (that is, the result is the first argument, no matter what the second argument is) is associative but not commutative. Likewise, the trivial operation <math>x \circ y = y</math> (that is, the result is the second argument, no matter what the first argument is) is associative but not commutative.
|4= Addition and multiplication of [[complex number]]s and [[quaternion]]s are associative. Addition of [[octonion]]s is also associative, but multiplication of octonions is non-associative.
|5= The [[greatest common divisor]] and [[least common multiple]] functions act associatively.

<math display="block">
\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}.
</math>
|6= Taking the [[intersection (set theory)|intersection]] or the [[union (set theory)|union]] of [[set (mathematics)|sets]]:
<math display="block">
\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.
</math>
|7= If {{mvar|M}} is some set and {{mvar|S}} denotes the set of all functions from {{mvar|M}} to {{mvar|M}}, then the operation of [[function composition]] on {{mvar|S}} is associative:<math display="block">(f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h\qquad\mbox{for all }f,g,h\in S.</math>
|8= Slightly more generally, given four sets {{mvar|M}}, {{mvar|N}}, {{mvar|P}} and {{mvar|Q}}, with {{math|{{var|h}} : {{var|M}} → {{var|N}}}}, {{math|{{var|g}} : {{var|N}} → {{var|P}}}}, and {{math|{{var|f}} : {{var|P}} → {{var|Q}}}}, then

<math display="block">(f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h</math>

as before. In short, composition of maps is always associative.
|9= In [[category theory]], composition of morphisms is associative by definition. Associativity of functors and natural transformations follows from associativity of morphisms.
|10= Consider a set with three elements, {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}. The following operation:

{{wikitable| class="wikitable" style="text-align:center"
|-
! × !! {{mvar|A}} !! {{mvar|B}} !! {{mvar|C}}
|-
! {{mvar|A}}
| {{mvar|A}} || {{mvar|A}} || {{mvar|A}}
|-
! {{mvar|B}}
| {{mvar|A}} || {{mvar|B}} || {{mvar|C}}
|-
! {{mvar|C}}
| {{mvar|A}} || {{mvar|A}} || {{mvar|A}}
}}

is associative. Thus, for example, {{math|1={{var|A}}({{var|B}}{{var|C}}) = ({{var|A}}{{var|B}}){{var|C}} = {{var|A}}}}. This operation is not commutative.
|11= Because [[Matrix (mathematics)|matrices]] represent [[linear map|linear function]]s, and [[matrix multiplication]] represents function composition, one can immediately conclude that matrix multiplication is associative.<ref>{{cite web|url=http://www.khanacademy.org/math/linear-algebra/matrix-transformations/composition-of-transformations/v/matrix-product-associativity|title=Matrix product associativity|publisher=Khan Academy|access-date=5 June 2016}}</ref>
|12= For [[real number]]s (and for any [[totally ordered set]]), the minimum and maximum operation is associative: <math display="block">\max(a, \max(b, c)) = \max(\max(a, b), c) \quad \text{ and } \quad \min(a, \min(b, c)) = \min(\min(a, b), c).</math>

}}

== Propositional logic ==
{{Transformation rules}}

=== Rule of replacement ===
In standard truth-functional propositional logic, ''association'',<ref>{{cite book |last1=Moore |first1=Brooke Noel |last2=Parker |first2=Richard |date=2017 |title=Critical Thinking |location=New York |publisher=McGraw-Hill Education |page=321 |isbn=9781259690877|edition=12th }}</ref><ref>{{cite book |last1=Copi |first1=Irving M. |last2=Cohen |first2=Carl |last3=McMahon |first3=Kenneth |date=2014 |title=Introduction to Logic |location=Essex |publisher=Pearson Education |page=387 |isbn=9781292024820|edition=14th }}</ref> or ''associativity''<ref>{{cite book |last1=Hurley |first1=Patrick J. |last2=Watson |first2=Lori |date=2016 |title=A Concise Introduction to Logic |location=Boston |publisher=Cengage Learning |page=427 |isbn=9781305958098|edition=13th }}</ref> are two [[Validity (logic)|valid]] [[rule of replacement|rules of replacement]]. The rules allow one to move parentheses in [[well-formed formula|logical expressions]] in [[formal proof|logical proofs]]. The rules (using [[Logical connective#In language|logical connectives]]  notation) are:

<math display="block">(P \lor (Q \lor R)) \Leftrightarrow ((P \lor Q) \lor R)</math>

and

<math display="block">(P \land (Q \land R)) \Leftrightarrow ((P \land Q) \land R),</math>

where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a [[Formal proof|proof]] with".

=== Truth functional connectives ===
''Associativity'' is a property of some [[logical connective]]s of truth-functional [[propositional logic]]. The following [[logical equivalence]]s demonstrate that associativity is a property of particular connectives. The following (and their converses, since {{math|↔}} is commutative) are truth-functional [[tautology (logic)|tautologies]].{{citation needed|reason=Stack Exchange is not a reliable source?|date=June 2022}}

;Associativity of disjunction
:<math>((P \lor Q) \lor R) \leftrightarrow (P \lor (Q \lor R))</math>
;Associativity of conjunction
:<math>((P \land Q) \land R) \leftrightarrow (P \land (Q \land R))</math>
;Associativity of equivalence
:<math>((P \leftrightarrow Q) \leftrightarrow R) \leftrightarrow (P \leftrightarrow (Q \leftrightarrow R))</math>

[[Logical NOR|Joint denial]] is an example of a truth functional connective that is ''not'' associative.

== Non-associative operation==
A binary operation <math>*</math> on a set ''S'' that does not satisfy the associative law is called '''non-associative'''. Symbolically,

<math display="block">(x*y)*z\ne x*(y*z)\qquad\mbox{for some }x,y,z\in S.</math>

For such an operation the order of evaluation ''does'' matter. For example:
; [[Subtraction]]
:<math>
(5-3)-2 \, \ne \, 5-(3-2)
</math>
; [[Division (mathematics)|Division]]
:<math>
(4/2)/2 \, \ne \, 4/(2/2)
</math>
; [[Exponentiation]]
:<math>
2^{(1^2)} \, \ne \, (2^1)^2
</math>
; [[Vector cross product]]
:<math>\begin{align}
  \mathbf{i} \times (\mathbf{i} \times \mathbf{j}) &= \mathbf{i} \times \mathbf{k} = -\mathbf{j} \\
  (\mathbf{i} \times \mathbf{i}) \times \mathbf{j} &= \mathbf{0} \times \mathbf{j} = \mathbf{0}
\end{align}</math>

Also although addition is associative for finite sums, it is not associative inside infinite sums ([[series (mathematics)|series]]). For example,
<math display="block">
(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+\dots =  0
</math>
whereas
<math display="block">
1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+\dots  =  1.
</math>

Some non-associative operations are fundamental in mathematics. They appear often as the multiplication in structures called [[non-associative algebra]]s, which have also an addition and a [[scalar multiplication]]. Examples are the [[octonion]]s and [[Lie algebra]]s. In Lie algebras, the multiplication satisfies [[Jacobi identity]] instead of the associative law; this allows abstracting the algebraic nature of [[infinitesimal transformation]]s.

Other examples are [[quasigroup]], [[quasifield]], [[non-associative ring]], and [[commutative non-associative magmas]].

=== Nonassociativity of floating-point calculation ===
In mathematics, addition and multiplication of real numbers are associative.  By contrast, in computer science, addition and multiplication of [[floating-point arithmetic|floating point numbers]] are ''not'' associative, as different rounding errors may be introduced when dissimilar-sized values are joined in a different order.<ref>Knuth, Donald, [[The Art of Computer Programming]], Volume 3, section 4.2.2</ref>

To illustrate this, consider a floating-point representation with a 4-bit [[significand]]:

{{block indent|1=(1.000<sub>2</sub>×2<sup>0</sup> + 1.000<sub>2</sub>×2<sup>0</sup>) +
1.000<sub>2</sub>×2<sup>4</sup> = 1.000<sub>2</sub>×2<sup>{{fontcolor|red|1}}</sup> + 1.000<sub>2</sub>×2<sup>4</sup> = 1.00{{fontcolor|red|1}}<sub>2</sub>×2<sup>4</sup>}}
{{block indent|1=1.000<sub>2</sub>×2<sup>0</sup> + (1.000<sub>2</sub>×2<sup>0</sup> +
1.000<sub>2</sub>×2<sup>4</sup>) = 1.000<sub>2</sub>×2<sup>{{fontcolor|red|0}}</sup> + 1.000<sub>2</sub>×2<sup>4</sup> = 1.00{{fontcolor|red|0}}<sub>2</sub>×2<sup>4</sup>}}

Even though most computers compute with 24 or 53 bits of significand,<ref>{{Cite book |title=IEEE Standard for Floating-Point Arithmetic |author=IEEE Computer Society |date=29 August 2008 |id=IEEE Std 754-2008|doi=10.1109/IEEESTD.2008.4610935 |ref=CITEREFIEEE_7542008 |isbn=978-0-7381-5753-5}}</ref> this is still an important source of rounding error, and approaches such as the [[Kahan summation algorithm]] are ways to minimize the errors. It can be especially problematic in parallel computing.<ref>{{Citation
| last1   = Villa
| first1   = Oreste
| last2   = Chavarría-mir
| first2   = Daniel
| last3   = Gurumoorthi
| first3   = Vidhya
| last4   = Márquez
| first4   = Andrés
| last5   = Krishnamoorthy
| first5   = Sriram
| title   = Effects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems
| url   = http://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdf
| access-date   = 8 April 2014
| archive-url   = https://web.archive.org/web/20130215171724/http://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdf
| archive-date   = 15 February 2013
| url-status   = dead
}}</ref><ref name="Goldberg_1991">{{cite journal|last=Goldberg|first=David|date=March 1991|title=What Every Computer Scientist Should Know About Floating-Point Arithmetic|url=http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf|journal=[[ACM Computing Surveys]]|volume=23|issue=1|pages=5–48|doi=10.1145/103162.103163|s2cid=222008826|access-date=20 January 2016|url-status=live|archive-url=https://web.archive.org/web/20220519083509/http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf|archive-date=2022-05-19}}</ref>

=== Notation for non-associative operations ===
{{main|Operator associativity}}

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 <math>\dfrac{2}{3/4}</math>). However, [[mathematician]]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.,

<math display="block">
\left.
\begin{array}{l}
a*b*c=(a*b)*c
\\
a*b*c*d=((a*b)*c)*d
\\
a*b*c*d*e=(((a*b)*c)*d)*e\quad
\\
\mbox{etc.}
\end{array}
\right\}
\mbox{for all }a,b,c,d,e\in S
</math>

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

<math display="block">
\left.
\begin{array}{l}
x*y*z=x*(y*z)
\\
w*x*y*z=w*(x*(y*z))\quad
\\
v*w*x*y*z=v*(w*(x*(y*z)))\quad\\
\mbox{etc.}
\end{array}
\right\}
\mbox{for all }z,y,x,w,v\in S
</math>

Both left-associative and right-associative operations occur. Left-associative operations include the following:
; Subtraction and division of real numbers<ref>George Mark Bergman [https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html "Order of arithmetic operations"]</ref><ref>[http://eduplace.com/math/mathsteps/4/a/index.html "The Order of Operations"]. Education Place.</ref><ref>[https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-order-of-operations/v/introduction-to-order-of-operations "The Order of Operations"], timestamp [https://www.youtube.com/watch?v=ClYdw4d4OmA&t=5m40s 5m40s]. [[Khan Academy]].</ref><ref>[http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3 "Using Order of Operations and Exploring Properties"] {{Webarchive|url=https://web.archive.org/web/20220716062834/http://www.doe.virginia.gov/instruction/mathematics/middle/algebra_readiness/curriculum_companion/order-operations.pdf#page=3 |date=2022-07-16 }}, section 9. Virginia Department of Education.</ref><ref name="Bronstein_1987">Bronstein, ''[[:de:Taschenbuch der Mathematik]]'', pages 115-120, chapter: 2.4.1.1, {{ISBN|978-3-8085-5673-3}}</ref>
:<math>x-y-z=(x-y)-z</math>
:<math>x/y/z=(x/y)/z</math>
; Function application
:<math>(f \, x \, y) = ((f \, x) \, y)</math>
This notation can be motivated by the [[currying]] isomorphism, which enables partial application.

Right-associative operations include the following:
; [[Exponentiation]] of real numbers in superscript notation
:<math>x^{y^z}=x^{(y^z)}</math><p>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:</p>
:<math>(x^y)^z=x^{(yz)}</math><p>Formatted correctly, the superscript inherently behaves as a set of parentheses; e.g. in the expression <math>2^{x+3}</math> the addition is performed [[order of operations|before]] the exponentiation despite there being no explicit parentheses <math>2^{(x+3)}</math> wrapped around it. Thus given an expression such as <math>x^{y^z}</math>, the full exponent <math>y^z</math> of the base <math>x</math> is evaluated first. However, in some contexts, especially in handwriting, the difference between <math>{x^y}^z=(x^y)^z</math>, <math>x^{yz}=x^{(yz)}</math> and <math>x^{y^z}=x^{(y^z)}</math> can be hard to see. In such a case, right-associativity is usually implied.</p>
; [[Function (mathematics)|Function definition]]
:<math>\mathbb{Z} \rarr \mathbb{Z} \rarr \mathbb{Z} = \mathbb{Z} \rarr (\mathbb{Z} \rarr \mathbb{Z})</math>
:<math>x \mapsto y \mapsto x - y = x \mapsto (y \mapsto x - y)</math><p>Using right-associative notation for these operations can be motivated by the [[Curry–Howard correspondence]] and by the [[currying]] isomorphism.</p>

Non-associative operations for which no conventional evaluation order is defined include the following.
; Exponentiation of real numbers in infix notation<ref name="Codeplea_2016">[https://codeplea.com/exponentiation-associativity-options Exponentiation Associativity and Standard Math Notation] Codeplea. 23 August 2016. Retrieved 20 September 2016.</ref>
:<math>(x^\wedge y)^\wedge z\ne x^\wedge(y^\wedge z)</math>
; [[Knuth's up-arrow notation|Knuth's up-arrow operators]]
:<math> a \uparrow \uparrow (b \uparrow \uparrow c) \ne (a \uparrow \uparrow b) \uparrow \uparrow c</math>
:<math> a \uparrow \uparrow \uparrow (b \uparrow \uparrow \uparrow c) \ne (a \uparrow \uparrow \uparrow b) \uparrow \uparrow \uparrow c</math>
; Taking the [[cross product]] of three vectors
:<math>\vec a \times (\vec b \times \vec c) \neq (\vec a \times \vec b ) \times \vec c \qquad \mbox{ for some } \vec a,\vec b,\vec c \in \mathbb{R}^3</math>
; Taking the pairwise [[average]] of real numbers
:<math>{(x+y)/2+z\over2}\ne{x+(y+z)/2\over2} \qquad \mbox{for all }x,y,z\in\mathbb{R} \mbox{ with }x\ne z.</math>
; Taking the [[complement (set theory)|relative complement]] of sets 
:<math>(A\backslash B)\backslash C \neq A\backslash (B\backslash C)</math>.<p>(Compare [[material nonimplication]] in logic.)</p>

== History ==

[[William Rowan Hamilton]] seems to have coined the term "associative property"<ref name="Hamilton">{{cite journal |author-link=William Rowan Hamilton |first=W.R. |last=Hamilton |year=1844–1850 |url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/ |title=On quaternions or a new system of imaginaries in algebra |journal=[[Philosophical Magazine]] |department=David R. Wilkins collection |publisher=[[Trinity College Dublin]]}}</ref> around 1844, a time when he was contemplating the non-associative algebra of the [[octonions]] he had learned about from [[John T. Graves]].<ref name="Baez">{{Cite journal | last1 = Baez | first1 = John C. | author-link = John Baez| title = The Octonions | journal = Bulletin of the American Mathematical Society | issn = 0273-0979 | volume = 39 | issue = 2 | pages = 145–205 | year = 2002 | url = https://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | doi = 10.1090/S0273-0979-01-00934-X | arxiv = math/0105155| mr = 1886087| s2cid = 586512}}</ref>

== Relationship with commutativity in certain special cases ==
In general, associative operations are not commutative.  However, under certain special conditions, it may be the case that associativity implies commutativity.  Associative operators defined on an interval of the real number line are commutative if they are continuous and injective in both arguments.<ref>{{Cite book |last=Aczél |first=J. |url=https://www.google.com/books/edition/Lectures_on_Functional_Equations_and_The/n7vckU_1tY4C |title=Lectures on Functional Equations and Their Applications |date=1966-01-01 |publisher=Academic Press |isbn=978-0-08-095525-4 |pages=267 |language=en |ol=46920179M |ol-access=free}}</ref> A consequence is that every continuous, associative operator on two real inputs that is strictly increasing in each of its inputs is commutative.<ref>{{Cite journal |last=Ling |first=Cho-Hsin |date=1 September 1964 |title=Representation of associative functions |url=https://scispace.com/pdf/representation-of-associative-functions-3chz5py4.pdf |journal=Publicationes Mathematicae |volume=12 |pages=189–212}}</ref>

==See also==
{{Wiktionary}}
* [[Light's associativity test]]
* [[Telescoping series]], the use of addition associativity for cancelling terms in an infinite [[series (mathematics)|series]]
* A [[semigroup]] is a set with an associative binary operation.
* [[Commutativity]] and [[distributivity]] are two other frequently discussed properties of binary operations.
* [[Power associativity]], [[alternativity]], [[flexible algebra|flexibility]] and [[N-ary associativity]] are weak forms of associativity.
* [[Moufang loop|Moufang identities]] also provide a weak form of associativity.

==References==
{{reflist}}

[[Category:Properties of binary operations]]
[[Category:Elementary algebra]]
[[Category:Functional analysis]]
[[Category:Rules of inference]]