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Associativity is a property of binary operations, which is a mathematical operation that takes two inputs and produces a single output. A binary operation is associative if the order in which the operations are performed does not affect the result. In other words, for any elements a, b, and c in a set, the operation * is associative if (a * b) * c = a * (b * c). This property is crucial in various areas of mathematics, including algebra, number theory, and abstract algebra.

Associativity is often contrasted with commutativity, another property of binary operations. While associativity concerns the grouping of operations, commutativity concerns the order of the operands. A binary operation is commutative if a * b = b * a for all elements a and b in the set.

Associativity is a fundamental concept in the study of algebraic structures, such as groups, rings, and fields. In these structures, the binary operation is typically associative, allowing for the definition of products and sums over multiple elements without ambiguity. For example, in a group, the group operation is associative, enabling the definition of the product of any finite number of group elements.

The concept of associativity can be extended to operations with more than two inputs, such as ternary operations. However, the definition of associativity for such operations is more complex and less commonly used.