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. , , );
- combine relations with the set operations — union, intersection, difference, complement — and form the inverse ;
- 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 by exercising its definitions on concrete problems. All theory needed is restated, self-contained, in 2method.md.