# 6 | Relations

Let $$R$$ be a relation on a set $$A$$, i.e., a subset of $$A \times A$$. Notation:
$$x R y \; \text{iff} \; (x,y) \in R \subseteq A \times A.$$
1. $$R$$ is reflexive iff $$x R x \; \forall x \in A.$$
2. $$R$$ is symmetric iff $$x R y \Rightarrow y R x \; \forall x,y \in A.$$
3. $$R$$ is antisymmetric iff $$x R y \wedge y R x \Rightarrow x = y \; \forall x,y \in A.$$
4. $$R$$ is transitive iff $$x R y \wedge y R z \Rightarrow x R z \; \forall x,y,z \in A.$$