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In mathematics, correspondence is an alternate term for a relation between two sets. In other words, a correspondence of sets X and Y is a subset of the Cartesian product XxY of the sets.

A one-to-one correspondence is another name for a bijection.

In economics, a correspondence between the sets A and B is usually thought of as a map f:AP(B)from the elements of a set A into the set of all subsets of a set B (the power set of B). This is equivalent to the first definition. However, there is usually an additional property that for all a in A, f(a) is not empty. In other words, each element in A maps to a non-empty subset of B; or in terms of a relation R as subset of AxB, R projects to A surjectively. With this additional property, a correspondence is thought of as the generalization of a function, rather than as a special case of a relation.

An example of a correspondence in economics is the best response correspondence in game theory, which gives the optimal action for a player as a function of the strategies of all other players. If there is always a unique best action given what the other players are doing, then this is a function. If for some opponent's strategy, there is a set of best responses that are equally good, then this is a correspondence.

In algebraic geometry a correspondence between algebraic varieties V and W is in the same fashion a subset R of VxW, which is in addition required to be closed in the Zariski topology. It therefore means any relation that is defined by algebraic equations. There are some important examples, even when V and W are algebraic curves: for example the Hecke operators of modular form theory may be considered as correspondences of modular curves.

Correspondence is also the communication between two or more correspondents, for instance, by letters that pass between the writers through a postal service.