, the associative property is a property of some binary operation
s, which means that rearranging the parentheses in an expression will not change the result. In propositional logic
, associativity is a valid rule of replacement
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
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:
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 operand
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
and matrix multiplication
are associative, but (generally) not commutative.
Associative operations are abundant in mathematics; in fact, many algebraic structure
s (such as semigroups
) explicitly require their binary operations to be associative.
However, many important and interesting operations are non-associative; some examples include subtraction
, 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.
Formally, a binary operation
∗ 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
) as for multiplication
:(''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
thumb|250px|In the absence of the associative property, five factors ''a, b, c, d, 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. 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:
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
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 A
C) is equivalent to (A
C, but A
C most commonly means (A
B and B
C), which is not equivalent.
thumb|The addition of real numbers is associative.
Some examples of associative operations include the following.
* The concatenation
of the three strings
can be computed by concatenating the first two strings (giving
) and appending the third string (
), or by joining the second and third string (giving
) and concatenating the first string (
) with the result. The two methods produce the same result; string concatenation is associative (but not commutative).
* In arithmetic
of real number
s are associative; i.e.,
: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
s and quaternion
s are associative. Addition of octonion
s is also associative, but multiplication of octonions is non-associative.
* The greatest common divisor
and least common multiple
functions act associatively.
* Taking the intersection
or the union
* If ''M'' is some set and ''S'' denotes the set of all functions from ''M'' to ''M'', then the operation of function composition
on ''S'' is associative:
* 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
: 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
represent linear function
s, and matrix multiplication
represents function composition, one can immediately conclude that matrix multiplication is associative.
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
in logical proofs
. The rules (using logical connectives
" is a metalogic
representing "can be replaced in a proof
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 are truth-functional tautologies
Associativity of disjunction:
Associativity of conjunction:
Associativity of equivalence:
Joint denial is an example of a truth functional connective that is ''not'' associative.
A binary operation
on a set ''S'' that does not satisfy the associative law is called non-associative. Symbolically,
For such an operation the order of evaluation ''does'' matter. For example:
Also note that infinite sums are not generally associative, for example:
The study of non-associative structures arises from reasons somewhat different from the mainstream of classical algebra. One area within non-associative algebra
that has grown very large is that of Lie algebra
s. There the associative law is replaced by the Jacobi identity
. Lie algebras abstract the essential nature of infinitesimal transformation
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
. Other examples are quasigroup
, non-associative ring
, non-associative algebra
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
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
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
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
). 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.,
while a right-associative operation is conventionally evaluated from right to left:
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
[Bronstein: :de:Taschenbuch der Mathematik, pages 115-120, chapter: 18.104.22.168, ]
* Function application:
:This notation can be motivated by the currying
Right-associative operations include the following:
of real numbers in superscript notation:
: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:
:Formatted correctly, the superscript inherently behaves as a set of parentheses; e.g. in the expression
the addition is performed before
the exponentiation despite there being no explicit parentheses
wrapped around it. Thus given an expression such as
, the full exponent
of the base
is evaluated first. However, in some contexts, especially in handwriting, the difference between
can be hard to see. In such a case, right-associativity is usually implied.
* Function definition
:Using right-associative notation for these operations can be motivated by the Curry–Howard correspondence
and by the currying
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](_blank)
Codeplea. 23 August 2016. Retrieved 20 September 2016.
* Knuth's up-arrow operators
* Taking the cross product
of three vectors:
* Taking the pairwise average
of real numbers:
* Taking the relative complement
is not the same as
. (Compare material nonimplication
* 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.
are two other frequently discussed properties of binary operations.
* Power associativity
and N-ary associativity
are weak forms of associativity.
* Moufang identities
also provide a weak form of associativity.
Category:Properties of binary operations
Category:Rules of inference