assoziativ

Assoziativ describes a property of binary operations where the grouping of operands does not affect the result. In mathematics and logic, an operation * on a set S is associative if for all a, b, c in S, (a*b)*c = a*(b*c). This allows expressions with multiple applications of the operation to be regrouped without changing the outcome.

Common examples include addition and multiplication of numbers, set union and intersection, and function composition (f∘g)∘h = f∘(g∘h). Matrix multiplication is also associative. In many contexts, associativity is assumed and is essential for notational simplicity and for defining algebraic structures such as semigroups and monoids.

Non-examples help illustrate the concept: subtraction and division are not associative, since (5−3)−2 ≠ 5−(3−2). Exponentiation is not associative in general, as (a^b)^c ≠ a^(b^c). In boolean logic, operators like AND and OR are associative as well.

In programming languages, associativity determines how expressions with multiple operators are parsed when parentheses are absent; most languages define either left- or right-associative evaluation for common operators. The term is frequently discussed alongside commutativity, which asks whether a*b = b*a. The associative property underpins many areas of mathematics and computer science, influencing the design of algorithms and the definition of algebraic structures.

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