order of the factors, in five 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 (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, 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 (using logical connectives notation) 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 (and their converses, since is commutative) 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 (series). 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 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