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

    E 6 Exercises