Associative Operations: Tuples and Sets

This lecture explores the concept of associative operations on tuples, demonstrating how combining associative functions results in another associative function. Examples include rational multiplication and computing averages efficiently. The lecture also discusses how commutativity and symmetry can lead to associativity, illustrated through examples of modular fractions and relativistic velocity addition. Furthermore, it delves into the consequences of non-associativity in floating-point arithmetic and presents a family of associative operations on sets, proving their associativity properties. The lecture concludes with detailed proofs and examples showcasing the properties of associative operations.