Editing Binary relation (section)

== Calculus of relations ==
Developments in [[algebraic logic]] have facilitated usage of binary relations. The [[calculus of relations]] includes the [[algebra of sets]], extended by [[composition of relations]] and the use of [[converse relation]]s. The inclusion <math>R \subseteq S,</math> meaning that <math>aRb</math> implies <math>aSb</math>, sets the scene in a [[Lattice (order theory)|lattice]] of relations. But since <math>P \subseteq Q \equiv (P \cap \bar{Q} = \varnothing ) \equiv (P \cap Q = P),</math> the inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according to [[composition of relations#Schröder rules|Schröder rules]], provides a calculus to work in the [[power set]] of <math>A \times B.</math>

In contrast to homogeneous relations, the [[composition of relations]] operation is only a [[partial function]]. The necessity of matching target to source of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of [[category theory]] as in the [[category of sets]], except that the [[morphism]]s of this category are relations. The {{em|objects}} of the category [[Category of relations|Rel]] are sets, and the relation-morphisms compose as required in a [[Category (mathematics)|category]].{{citation needed|reason=Who has suggested this, when, and where?|date=June 2021}}