Relation

A relation is any subset of a Cartesian product. For instance, a relation \(R\) over any subset of \(X \times X\) is called a "Relation on \(X\)".

• A Relation \(R\) is said to be reflexive if \(\forall x \in X: (x,x) \in R\).
• A Relation \(R\) is said to be symmetric if \(\forall x,y \in X: (x,y) \in R \implies (y,x) \in R\).
• A Relation \(R\) is antisymmetric if \(\forall x,y \in X: (x,y) \in R \land (y,x) \in R \implies x = y\).
• A Relation \(R\) is transitive if \(\forall x,y,z \in X: (x,y) \in R \land (y,z) \in R \implies (x,z) \in R\).
• A Relation \(R\) is an equivalence relation if it is reflexive, symmetric and transitive.