<Li> A × B = B × A = A × B . </Li> <P> Set theory is seen as the foundation from which virtually all of mathematics can be derived . For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations . </P> <P> One of the main applications of naive set theory is constructing relations . A relation from a domain A to a codomain B is a subset of the Cartesian product A × B. Given this concept, we are quick to see that the set F of all ordered pairs (x, x), where x is real, is quite familiar . It has a domain set R and a codomain set that is also R, because the set of all squares is subset of the set of all reals . If placed in functional notation, this relation becomes f (x) = x . The reason these two are equivalent is for any given value, y that the function is defined for, its corresponding ordered pair, (y, y) is a member of the set F . </P> <P> Although initially naive set theory, which defines a set merely as any well - defined collection, was well accepted, it soon ran into several obstacles . It was found that this definition spawned several paradoxes, most notably: </P>

Difference between well defined set and not well defined set