Associative property
In mathematics, the associative property is a property of some binary operations 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 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 operands is not changed. That is, 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,, 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 commutative.
Associative operations are abundant in mathematics; in fact, many algebraic structures 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
Formally, a binary operation on a set is called associative if it satisfies the associative law:Here, ∗ is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbol as for multiplication.
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.The number of possible bracketings is just the Catalan number,
, for n operations on n+1 values. For instance, a product of 3 operations on 4 elements may be written, in possible ways:
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, 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, is equivalent to, but most commonly means, which is not equivalent.
Examples
Some examples of associative operations include the following.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 in logical proofs. The rules are:and
where "" is a metalogical symbol representing "can be replaced in a proof with".
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 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.
Non-associative operation
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:
; Subtraction
; Division
; Exponentiation
; Vector cross product
Also although addition is associative for finite sums, it is not associative inside infinite sums. For example,
whereas
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 octonions and Lie algebras. 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.
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.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 minimize 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 evaluation if a non-associative operation appears more than once in an expression. 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.,
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
; Function application
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
; Function definition
Non-associative operations for which no conventional evaluation order is defined include the following.
; Exponentiation of real numbers in infix notation
; Knuth's up-arrow operators
; Taking the cross product of three vectors
; Taking the pairwise average of real numbers
; Taking the relative complement of sets