---
title: "Central ideas of formal and informal proofs"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:formal-informal-proofs-central-ideas"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/formal-informal-proofs-central-ideas
site: BCT Engineering
---

# Central ideas of formal and informal proofs

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2073 Magh, 4 marks
- Video: [An Introduction to Mathematical Proofs](https://www.youtube.com/watch?v=YYgepDY3rHw) — zeropercent
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/formal-informal-proofs-central-ideas
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

What are the central ideas of formal and informal proofs?

## Short answer

Formal proofs are symbolic, step-by-step sequences using explicit inference rules, machine-checkable. Informal proofs use natural language, skip routine steps, and rely on human judgment to convince that a formal proof exists.

## Answer

#### Formal proof
A formal proof is a finite sequence of statements written in a precise formal language. Each statement is an axiom, a hypothesis (premise), or follows from earlier statements by a stated rule of inference (modus ponens, modus tollens, resolution and so on). No step is skipped and no meaning is needed to check it, so a computer can verify it. Example: the step-by-step proofs with a reason for each line.

**Central idea:** correctness is purely mechanical and symbolic, with complete rigour.

#### Informal proof
An informal proof is an argument written in ordinary language mixed with mathematics. Routine steps are skipped, known theorems are used freely, and the reader is expected to fill gaps. Examples: direct proof, proof by contradiction, proof by cases as in textbooks.

**Central idea:** to convince a human reader that a formal proof could be written, using readable and shorter reasoning.

#### Comparison
| Point | Formal proof | Informal proof |
|---|---|---|
| Language | Symbolic logic | Natural language plus maths |
| Steps | Every step shown with a rule | Obvious steps omitted |
| Checking | Can be machine-checked | Checked by human judgement |
| Length | Long | Short |
| Use | Verification, theorem provers | Textbooks, everyday mathematics |
| Risk | Very low | Hidden gaps possible |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the central idea of a formal proof?
   A. Mechanical, symbolic correctness with every step justified by a rule
   B. Convincing a reader quickly
   C. Skipping obvious steps
   D. Using natural language
   Answer: A. Each step is an axiom, hypothesis or follows by a stated rule, so a computer can check it. (Key point: Formal proof)

2. What is the central idea of an informal proof?
   A. To convince a human reader that a formal proof could be written
   B. To be checked by a machine
   C. To list every inference rule
   D. To avoid mathematics
   Answer: A. It uses readable, shorter reasoning with gaps left to the reader. (Key point: Informal proof)

3. Which is true of formal proofs compared with informal ones?
   A. Shorter
   B. Can be machine-checked
   C. Use ordinary language
   D. Risk of hidden gaps is high
   Answer: B. Formal proofs are long but machine-checkable. (Key point: Comparison)

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