Associative property

 ( Properties Of Binary Operations ) 

In mathematics, the associative property

is a property of some that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in Formal proof.

Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the 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:

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 , 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, , 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 structures (such as semigroups and 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.

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Definition Formally, a binary operation $\backslash ast$ on a set is called associative if it satisfies the associative law:

$(x\; \backslash ast\; y)\; \backslash ast\; z\; =\; x\; \backslash ast\; (y\; \backslash ast\; z)$, for all $x,y,z$ in .

Here, ∗ is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbol (juxtaposition) as for multiplication.

$(xy)z\; =\; x(yz)$, for all $x,y,z$ in .

The associative law can also be expressed in functional notation thus: $(f\; \backslash circ\; (g\; \backslash circ\; h))(x)\; =\; ((f\; \backslash circ\; g)\; \backslash circ\; h)(x)$

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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.

John R. Durbin (1992). 9780471510017, Wiley. . ISBN 9780471510017

This is called the generalized associative law.

The number of possible bracketings is just the Catalan number, $C\_n$ , 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 $C\_3\; =\; 5$ possible ways:

• $((ab)c)d$
• $(a(bc))d$
• $a((bc)d)$
• $a(b(cd))$
• $(ab)(cd)$

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

$abcd$

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 . It is associative; thus, is equivalent to , but most commonly means , which is not equivalent.

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Examples Some examples of associative operations include the following.

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Propositional logic

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Rule of replacement In standard truth-functional propositional logic, association,

(2025). 9781259690877, McGraw-Hill Education. ISBN 9781259690877

(2025). 9781292024820, Pearson Education. ISBN 9781292024820

or associativity

(2025). 9781305958098, Cengage Learning. ISBN 9781305958098

are two valid rules of replacement. The rules allow one to move parentheses in logical expressions in formal proof. The rules (using logical connectives notation) are:

$$(P\; \backslash lor\; (Q\; \backslash lor\; R))\; \backslash Leftrightarrow\; ((P\; \backslash lor\; Q)\; \backslash lor\; R)$$

and

$$(P\; \backslash land\; (Q\; \backslash land\; R))\; \backslash Leftrightarrow\; ((P\; \backslash land\; Q)\; \backslash land\; R),$$

where "$\backslash Leftrightarrow$" is a symbol representing "can be replaced in a Formal proof with".

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Truth functional connectives Associativity is a property of some logical connectives of truth-functional propositional logic. The following logical equivalences demonstrate that associativity is a property of particular connectives. The following (and their converses, since is commutative) are truth-functional tautologies.

Associativity of disjunction

$((P\; \backslash lor\; Q)\; \backslash lor\; R)\; \backslash leftrightarrow\; (P\; \backslash lor\; (Q\; \backslash lor\; R))$

Associativity of conjunction

$((P\; \backslash land\; Q)\; \backslash land\; R)\; \backslash leftrightarrow\; (P\; \backslash land\; (Q\; \backslash land\; R))$

Associativity of equivalence

$((P\; \backslash leftrightarrow\; Q)\; \backslash leftrightarrow\; R)\; \backslash leftrightarrow\; (P\; \backslash leftrightarrow\; (Q\; \backslash leftrightarrow\; R))$

Logical NOR is an example of a truth functional connective that is not associative.

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

$$(x*y)*z\backslash ne\; x*(y*z)\backslash qquad\backslash mbox\{for\; some\; \}x,y,z\backslash in\; S.$$

For such an operation the order of evaluation does matter. For example:

Subtraction

$$

(5-3)-2 \, \ne \, 5-(3-2)

Division

$$

(4/2)/2 \, \ne \, 4/(2/2)

Exponentiation

$$

2^{(1^2)} \, \ne \, (2^1)^2

Vector cross product

$\backslash begin\{align\}$

 \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}

Also although addition is associative for finite sums, it is not associative inside infinite sums (series). For example, $$(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+\backslash dots\; =\; 0$$ whereas $$1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+\backslash dots\; =\; 1.$$

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

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

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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 numbers are not associative, as different rounding errors may be introduced when dissimilar-sized values are joined in a different order.Knuth, Donald, The Art of Computer Programming, Volume 3, section 4.2.2

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

Even though most computers compute with 24 or 53 bits of significand,

this is still an important source of rounding error, and approaches such as the Kahan summation algorithm are ways to minimise the errors. It can be especially problematic in parallel computing.

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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 (unless the notation specifies the order in another way, like $\backslash dfrac\{2\}\{3/4\}$). However, 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.,

$$\backslash left.\; \backslash begin\{array\}\{l\}\; a*b*c=(a*b)*c\; \backslash \backslash \; a*b*c*d=((a*b)*c)*d\; \backslash \backslash \; a*b*c*d*e=(((a*b)*c)*d)*e\backslash quad\; \backslash \backslash \; \backslash mbox\{etc.\}\; \backslash end\{array\}\; \backslash right\backslash \}\; \backslash mbox\{for\; all\; \}a,b,c,d,e\backslash in\; S$$

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

$$\backslash left.\; \backslash begin\{array\}\{l\}\; x*y*z=x*(y*z)\; \backslash \backslash \; w*x*y*z=w*(x*(y*z))\backslash quad\; \backslash \backslash \; v*w*x*y*z=v*(w*(x*(y*z)))\backslash quad\backslash \backslash \; \backslash mbox\{etc.\}\; \backslash end\{array\}\; \backslash right\backslash \}\; \backslash mbox\{for\; all\; \}z,y,x,w,v\backslash in\; S$$

Both left-associative and right-associative operations occur. Left-associative operations include the following:

$x-y-z=(x-y)-z$

$x/y/z=(x/y)/z$

Function application

$(f\; \backslash ,\; x\; \backslash ,\; y)\; =\; ((f\; \backslash ,\; x)\; \backslash ,\; y)$

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

$x^\{y^z\}=x^\{(y^z)\}$

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:

$(x^y)^z=x^\{(yz)\}$

Formatted correctly, the superscript inherently behaves as a set of parentheses; e.g. in the expression $2^\{x+3\}$ the addition is performed before the exponentiation despite there being no explicit parentheses $2^\{(x+3)\}$ wrapped around it. Thus given an expression such as $x^\{y^z\}$, the full exponent $y^z$ of the base $x$ is evaluated first. However, in some contexts, especially in handwriting, the difference between $\{x^y\}^z=(x^y)^z$, $x^\{yz\}=x^\{(yz)\}$ and $x^\{y^z\}=x^\{(y^z)\}$ can be hard to see. In such a case, right-associativity is usually implied.

Function definition

$\backslash mathbb\{Z\}\; \backslash rarr\; \backslash mathbb\{Z\}\; \backslash rarr\; \backslash mathbb\{Z\}\; =\; \backslash mathbb\{Z\}\; \backslash rarr\; (\backslash mathbb\{Z\}\; \backslash rarr\; \backslash mathbb\{Z\})$

$x\; \backslash mapsto\; y\; \backslash mapsto\; x\; -\; y\; =\; x\; \backslash mapsto\; (y\; \backslash mapsto\; x\; -\; y)$

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.

$(x^\backslash wedge\; y)^\backslash wedge\; z\backslash ne\; x^\backslash wedge(y^\backslash wedge\; z)$

Knuth's up-arrow operators

$a\; \backslash uparrow\; \backslash uparrow\; (b\; \backslash uparrow\; \backslash uparrow\; c)\; \backslash ne\; (a\; \backslash uparrow\; \backslash uparrow\; b)\; \backslash uparrow\; \backslash uparrow\; c$

$a\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; (b\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; c)\; \backslash ne\; (a\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; b)\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; c$

Taking the cross product of three vectors

$\backslash vec\; a\; \backslash times\; (\backslash vec\; b\; \backslash times\; \backslash vec\; c)\; \backslash neq\; (\backslash vec\; a\; \backslash times\; \backslash vec\; b\; )\; \backslash times\; \backslash vec\; c\; \backslash qquad\; \backslash mbox\{\; for\; some\; \}\; \backslash vec\; a,\backslash vec\; b,\backslash vec\; c\; \backslash in\; \backslash mathbb\{R\}^3$

Taking the pairwise average of real numbers

$\{(x+y)/2+z\backslash over2\}\backslash ne\{x+(y+z)/2\backslash over2\}\; \backslash qquad\; \backslash mbox\{for\; all\; \}x,y,z\backslash in\backslash mathbb\{R\}\; \backslash mbox\{\; with\; \}x\backslash ne\; z.$

Taking the relative complement of sets

$(A\backslash backslash\; B)\backslash backslash\; C\; \backslash neq\; A\backslash backslash\; (B\backslash backslash\; C)$.

(Compare material nonimplication in logic.)

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History William Rowan Hamilton seems to have coined the term "associative property" around 1844, a time when he was contemplating the non-associative algebra of the octonions he had learned about from John T. Graves.

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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.

J. Aczél (1966). 9780080955254, Academic Press. . ISBN 9780080955254

A consequence is that every continuous, associative operator on two real inputs that is strictly increasing in each of its inputs is commutative.

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See also

• Light's associativity test
• Telescoping series, the use of addition associativity for cancelling terms in an infinite 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 and N-ary associativity are weak forms of associativity.
• Moufang loop also provide a weak form of associativity.

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