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 operands 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 numbers, 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, a × b = b × 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 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.
Definition
edit
Formally, a binary operation∗ on a setS 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: f(f(x, y), z) = f(x, f(y, z)).
Generalized associative law
edit
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.^{[2]} 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
(ab)(cd)
(a(bc))d
a((bc)d)
a(b(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
An example where this does not work is the logical biconditional↔. It is associative; thus, A ↔ (B ↔ C) is equivalent to (A ↔ B) ↔ C, but A ↔ B ↔ C most commonly means (A ↔ B) and (B ↔ C), which is not equivalent.
Examples
edit
Some examples of associative operations include the following.
The concatenation 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).
Because of associativity, the grouping parentheses can be omitted without ambiguity.
The trivial operation x ∗ y = 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 x ∘ y = y (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 numbers and quaternions are associative. Addition of octonions is also associative, but multiplication of octonions is non-associative.
as before. In short, composition of maps is always associative.
In category theory, composition of morphisms is associative by definition. Associativity of functors and natural transformations follows from associativity of morphisms.
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 = A. This operation is not commutative.
Because matrices represent linear functions, and matrix multiplication represents function composition, one can immediately conclude that matrix multiplication is associative.^{[3]}
where "$\Leftrightarrow$" is a metalogicalsymbol representing "can be replaced in a proof with".
Truth functional connectives
edit
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.^{[citation needed]}
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 different rounding errors may be introduced when dissimilar-sized values are joined together in a different order.^{[7]}
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,^{[8]} 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.^{[9]}^{[10]}
Notation for non-associative operations
edit
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 ${\dfrac {2}{3/4}}$). However, mathematicians 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.,
$\left.{\begin{array}{l}a*b*c=(a*b)*c\\a*b*c*d=((a*b)*c)*d\\a*b*c*d*e=(((a*b)*c)*d)*e\quad \\{\mbox{etc.}}\end{array}}\right\}{\mbox{for all }}a,b,c,d,e\in S$
while a right-associative operation is conventionally evaluated from right to left:
$\left.{\begin{array}{l}x*y*z=x*(y*z)\\w*x*y*z=w*(x*(y*z))\quad \\v*w*x*y*z=v*(w*(x*(y*z)))\quad \\{\mbox{etc.}}\end{array}}\right\}{\mbox{for all }}z,y,x,w,v\in S$
Both left-associative and right-associative operations occur. Left-associative operations include the following:
Subtraction and division of real numbers^{[11]}^{[12]}^{[13]}^{[14]}^{[15]}
$x-y-z=(x-y)-z$
$x/y/z=(x/y)/z$
Function application
$(f\,x\,y)=((f\,x)\,y)$
This notation can be motivated by the currying isomorphism, which enables partial application.
Right-associative operations include the following:
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.
William Rowan Hamilton seems to have coined the term "associative property"^{[17]} around 1844, a time when he was contemplating the non-associative algebra of the octonions he had learned about from John T. Graves.^{[18]}
See also
edit
Look up associative property in Wiktionary, the free dictionary.
^
Hungerford, Thomas W. (1974). Algebra (1st ed.). Springer. p. 24. ISBN 978-0387905181. Definition 1.1 (i) a(bc) = (ab)c for all a, b, c in G.
^Durbin, John R. (1992). Modern Algebra: an Introduction (3rd ed.). New York: Wiley. p. 78. ISBN 978-0-471-51001-7. If $a_{1},a_{2},\dots ,a_{n}\,\,(n\geq 2)$ are elements of a set with an associative operation, then the product $a_{1}a_{2}\cdots a_{n}$ is unambiguous; this is, the same element will be obtained regardless of how parentheses are inserted in the product.
^"Matrix product associativity". Khan Academy. Retrieved 5 June 2016.
^Moore, Brooke Noel; Parker, Richard (2017). Critical Thinking (12th ed.). New York: McGraw-Hill Education. p. 321. ISBN 9781259690877.
^Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2014). Introduction to Logic (14th ed.). Essex: Pearson Education. p. 387. ISBN 9781292024820.
^Hurley, Patrick J.; Watson, Lori (2016). A Concise Introduction to Logic (13th ed.). Boston: Cengage Learning. p. 427. ISBN 9781305958098.
^IEEE Computer Society (29 August 2008). IEEE Standard for Floating-Point Arithmetic. doi:10.1109/IEEESTD.2008.4610935. ISBN 978-0-7381-5753-5. IEEE Std 754-2008.
^Villa, Oreste; Chavarría-mir, Daniel; Gurumoorthi, Vidhya; Márquez, Andrés; Krishnamoorthy, Sriram, Effects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems(PDF), archived from the original (PDF) on 15 February 2013, retrieved 8 April 2014
^Goldberg, David (March 1991). "What Every Computer Scientist Should Know About Floating-Point Arithmetic" (PDF). ACM Computing Surveys. 23 (1): 5–48. doi:10.1145/103162.103163. S2CID 222008826. Archived (PDF) from the original on 2022-05-19. Retrieved 20 January 2016.
^George Mark Bergman "Order of arithmetic operations"
^Baez, John C. (2002). "The Octonions" (PDF). Bulletin of the American Mathematical Society. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087. S2CID 586512.