# Laboratory Work 3 — Relations and Their Properties

> Third laboratory work of **Computer Discrete Mathematics**. You implement checks
> for the fundamental properties of a binary **relation on a set** — reflexivity,
> symmetry, transitivity, and antisymmetry — combine them to recognise an **equivalence
> relation** and a **partial order**, and generate the **inverse** of a relation.

## At a glance

| | |
|---|---|
| **Topic** | Binary relations on a set: reflexivity, symmetry, transitivity, antisymmetry, equivalence, partial order, inverse |
| **Prerequisite lectures** | [L03 — Relations](../../Lectures/CDM-L03.md) |
| **Deliverable** | Source code + report (see [5report.md](5report.md)) |
| **Grading** | Banded: Easy 60–74 / Medium 75–89 / Hard 90–100 (see [4task.md](4task.md)) |

## Contents

| # | Part | File |
|:--:|---|---|
| 1 | Objective | [1purpose.md](1purpose.md) |
| 2 | Methodical guidelines (theory & algorithms) | [2method.md](2method.md) |
| 4 | Task and order of execution | [4task.md](4task.md) |
| 5 | Report contents | [5report.md](5report.md) |
| 6 | Control questions | [6questions.md](6questions.md) |

## Conventions

- **Language** — source code and the report are in **English**.
- **Implement it yourself — no built-in shortcuts.** Do not call a library or
  built-in that performs the core task for you (e.g. no `HashSet`/`set()`/`Set`
  for sets, no library graph/relation type, no `math.gcd`, no `eval`). Build the
  mechanism from basic primitives — arrays/lists, loops, arithmetic, strings.
- **Deliverable** — submit the source code **and** a report structured as in
  [5report.md](5report.md); the code must reproduce the example outputs in
  [4task.md](4task.md).
- **Self-contained theory** — everything needed is in [2method.md](2method.md);
  no external material is required.

## Summary

A **relation on a set** is a set of ordered pairs drawn from $A \times A$, and its
character is captured by a few key properties. **Reflexivity**, **symmetry**, and
**transitivity** together define an **equivalence relation** — the abstraction
that partitions a set into classes of "equivalent" elements, one of the most
reused ideas in mathematics and computing. In this work you test each property in
isolation, combine them into an **equivalence** check and a **partial-order** check,
and construct the **inverse** of a relation. The tasks are grouped into three
difficulty bands.