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 \(x R y \wedge y R z \Rightarrow x R z \; \forall x,y,z \in A. \)