# Practical Work 2 — Relations: Properties, Representations, and Operations

> Second practical class of **Computer Discrete Mathematics**. Working **by hand**,
> you classify relations by the six key properties (reflexive, irreflexive,
> symmetric, antisymmetric, asymmetric, transitive), combine relations with set operations and the
> inverse, and represent a relation three ways — ordered pairs, a Boolean matrix,
> and a directed graph — to decide its **type**.

## At a glance

| | |
|---|---|
| **Topic** | Binary relations: properties, representations, operations, types |
| **Prerequisite lectures** | [L03 — Relations](../../Lectures/CDM-L03.md) (built on [L01](../../Lectures/CDM-L01.md)–[L02](../../Lectures/CDM-L02.md)) |
| **Mode** | In-class problem solving — pen and paper, no computer |
| **Deliverable** | Worked solutions to the tasks in [3classroom.md](3classroom.md) |

## Contents

| # | Part | File |
|:--:|---|---|
| 1 | Objective | [1purpose.md](1purpose.md) |
| 2 | Methodical guidelines (self-contained theory) | [2method.md](2method.md) |
| 3 | Classroom tasks **with solutions** | [3classroom.md](3classroom.md) |
| 6 | Control questions | [6questions.md](6questions.md) |

## Conventions

- **Language** — all work is written in **English**.
- **By hand, from the definitions.** Classify each relation directly against the
  definitions of the six properties, and justify every "yes" and every "no" —
  a "no" needs a **specific counterexample** (a witness pair).
- **Notation is fixed and self-contained.** The symbols $A \times B$, $a\,R\,b$,
  $(a,b)$, $R^{-1}$, and the property names are defined in
  [2method.md](2method.md); no outside reference is needed.
- **Three representations, one relation.** A relation may be written as a set of
  **ordered pairs**, a **Boolean matrix**, or a **directed graph**; the three are
  interchangeable, and a task may ask for any of them.
- **Every task has a full solution.** [3classroom.md](3classroom.md) gives each
  task together with a complete worked answer.

## Summary

Relations are how discrete mathematics models "is related to" — divisibility,
ordering, equality, adjacency, and the rows of a database table are all relations.
This practical drills the core skill: **testing a relation for the six
properties** and reading off its **type**. The tasks move from classifying six
small relations on $\{1,2,3,4\}$, through relations defined by a formula on the
integers, to combining relations with union/intersection/difference/complement
and the **inverse**, and finish by drawing one relation as pairs, a matrix, and a
graph to recognise it as a **partial order**. These are the exact skills used for
equivalence relations, orders, functions, and graphs in the rest of the course.