Correspondence (mathematics): Difference between revisions

Revision as of 02:46, 14 September 2005

Revision as of 02:54, 14 September 2005

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* In general [[mathematics]], '''correspondence''' is an alternate term for a [[mathematical relation|relation]] between two [[set]]s. Hence a correspondence of sets ''X'' and ''Y'' is any [[subset]] of the [[Cartesian product]] ''X''×''Y'' of the sets.

* In [[economics]], a '''correspondence''' between thetwo sets ''A'' and ''B'' is a [[map (mathematics)|map]] f:''A''→''P''(''B'') from the elements of athe set ''A'' to the subsets[[power ofset]] a setof ''B''. This is similar to a correspondence as defined in general mathematics (i.e., a [[mathematical relation|relation]],) except that the range is over sets instead of elements. However, there is usually the 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 ''A''×''B'', ''R'' projects to ''A'' [[surjective]]ly. A correspondence with this additional property is thought of as the generalization of a [[function (mathematics)|function]], rather than as a special case of a relation.

:An example of a correspondence in this sense 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.