Primary reference: Chapter 11 (Relations) of Book of Proof by Richard Hammack

richardhammack.github.io

Content covered in this unit includes:

• The definition of a relation on a set $X$.
• Properties of a relation on a set $X$:
  • Reflexive: $x \sim x$ for all $x \in X$.
  • Transitive: If $x \sim y$ and $y \sim z$, then $x \sim z$.
  • Symmetric: If $x \sim y$, then $y \sim x$.
  • Antisymmetric: If $x \sim y$ and $y \sim x$, then $x = y$.
• The definition of an equivalence relation: a relation which is reflexive, symmetric, and transitive.
• The definition of a partial order: a relation which is reflexive, antisymmetric, and transitive.
• Examples of equivalence relations (congruence modulo $n$) and partial orders (divisibility of natural numbers).
• The equivalence class of an element under a relation.