# 1. Objective / Мета роботи

**Implement checks for the fundamental properties of a binary relation on a set,
recognise equivalence relations, and generate the inverse of a relation.**

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

- represent a **binary relation on a set** $A$ as a set of ordered pairs
  $R \subseteq A \times A$;
- test the three core properties — **reflexivity** ($\forall a,\ (a,a)\in R$),
  **symmetry** ($(a,b)\in R \Rightarrow (b,a)\in R$), and **transitivity**
  ($(a,b),(b,c)\in R \Rightarrow (a,c)\in R$);
- combine them to decide whether $R$ is an **equivalence relation**, and explain
  how such a relation **partitions** $A$ into equivalence classes;
- construct the **inverse relation** $R^{-1} = \{(b,a) : (a,b)\in R\}$;
- test **antisymmetry** and decide whether $R$ is a **partial order** (reflexive,
  antisymmetric, and transitive);
- reason about the **time complexity** of each check.

The work makes the abstract properties of relations concrete and prepares the
ground for order relations, functions, and partitions studied later.