# Practical Work 4 — Minimizing Boolean Functions: Canonical Forms and Karnaugh Maps

> Fourth practical class of **Computer Discrete Mathematics**. Working **by hand**,
> you write a Boolean function's **canonical** forms (canonical DNF/CNF), reduce
> it to a **minimal** form both **algebraically** and with a **Karnaugh map**.

## At a glance

| | |
|---|---|
| **Topic** | Boolean minimization: canonical DNF/CNF, algebraic reduction, Karnaugh maps, minimal DNF/CNF |
| **Prerequisite lectures** | [L05 — Boolean algebra](../../Lectures/CDM-L05.md) (built on [L04](../../Lectures/CDM-L04.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.** Minimise using the laws of Boolean algebra and Karnaugh maps, not a
  computer-algebra tool; show the grouping on the map and the term each group
  yields.
- **Notation is fixed and self-contained.** An overline $\overline{x}$ means
  $\neg x$; **juxtaposition** $xy$ means $x \wedge y$; and $\vee$ (sometimes
  written $+$) is OR. The terms *literal, minterm, maxterm, canonical DNF/CNF,
  minimal DNF/CNF* are defined in [2method.md](2method.md); no outside reference is
  needed.
- **Every task has a full solution.** [3classroom.md](3classroom.md) gives each
  task together with a complete worked answer, including the Karnaugh map and the
  groups.

## Summary

A Boolean function can be written many ways; this practical is about finding a
**small** one. From a truth table we read the **canonical** DNF and CNF (a sum of
full minterms / a product of full maxterms), then shrink them — **algebraically**
with the laws (De Morgan, distributivity, combining, absorption) and **visually** with
a **Karnaugh map**, where adjacent $1$s combine into ever-larger groups that drop
variables. Fewer literals in the resulting minimal DNF/CNF mean a simpler circuit —
fewer gates, less power, and shorter delay — which is why minimization is a core
skill of logic design.