Associative property

In mathematics, the associative property is a property of some binary operations, which means 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, (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, , 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 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.
Formally, a binary operation ∗ on a set "S" 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 (juxtaposition) 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. 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 formula_9. It is associative, thus Aformula_9(Bformula_9C) is equivalent to (Aformula_9B)formula_9C, but Aformula_9Bformula_9C most commonly means (Aformula_9B and Bformula_9C), 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 (using logical connectives notation) are:
and
where "formula_25" 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 formula_32 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:
whereas
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 algebras. There the associative law is replaced by the Jacobi identity. Lie algebras abstract the essential nature of infinitesimal transformations, 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, quasifield, 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:
(1.0002×20 +
1.0002×20) +
1.0002×24 =
1.0002×2 +
1.0002×24 =
1.002×24
1.0002×20 +
(1.0002×20 +
1.0002×24) =
1.0002×2 +
1.0002×24 =
1.002×24
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 evaluation if a non-associative operation appears more than once in an expression (unless the notation specifies the order in another way, like formula_39). 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:
Right-associative operations include the following:
Non-associative operations for which no conventional evaluation order is defined include the following.