# 1. Objective / Мета роботи

**Gain fluency with binary relations — classifying them by their properties,
combining them, and moving freely between their three representations to
determine a relation's type.**

By the end of the class the student should be able to:

- **test** a relation for the core properties — **reflexive**, **irreflexive**,
  **symmetric**, **antisymmetric**, **asymmetric**, and **transitive** (and the
  rarer **antitransitive**) — and justify each verdict, supporting every "no" with
  a concrete **counterexample** (a witness pair);
- classify relations given **explicitly** (a list of pairs on a small set) and
  relations given by a **rule** on an infinite set such as the integers
  (e.g. $x \neq y$, $xy \ge 1$, $x = y^2$);
- **combine** relations with the set operations — union, intersection,
  difference, complement — and form the **inverse** $R^{-1}$;
- **represent** one relation three equivalent ways — as a set of **ordered
  pairs**, a **Boolean matrix**, and a **directed graph** — and translate between
  them;
- recognise a relation's **type**: an **equivalence relation** (reflexive +
  symmetric + transitive, giving a partition into classes) or a **partial order**
  (reflexive + antisymmetric + transitive).

The class consolidates
[Lecture 3 — Relations](../../Lectures/CDM-L03.md) by exercising its
definitions on concrete problems. All theory needed is restated, self-contained,
in [2method.md](2method.md).