# Correspondence

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:*A*→*P*(*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.