Proof — Basic

Level: Beginner

Basic techniques for writing proofs

What you will learn in this section

In the introductory section you acquired the skill of reading proofs. In this section you will learn the techniques for writing them.

  • How to write a direct proof
  • Proof by contradiction
  • Contrapositive proof
  • Mathematical induction

Contents

  1. Chapter 1 Direct proof

    The basic form: start from the assumptions and reach the conclusion.

  2. Chapter 2 Proof by contradiction

    Proving a statement by deriving a contradiction from its negation.

  3. Chapter 3 Contrapositive proof

    Proving the original statement by proving its contrapositive.

  4. Chapter 4 Mathematical induction

    A powerful method for proving statements about the natural numbers.

  5. Chapter 5 Existence and uniqueness proofs

    How to show that "there exists" and that "there is exactly one."

  6. Chapter 6 Exercises

    A summary of the basic section.

Prerequisites

  • The content of the introductory section (propositions, logical operations, implication, contrapositive)
  • Mathematics at the junior-high-school level

Frequently Asked Questions

What does the basic section on proof cover?

Five fundamental techniques: direct proof, proof by contradiction, contrapositive proof, mathematical induction, and proofs of existence and uniqueness. Through concrete worked examples, learners acquire the skill of writing proofs that proceed logically from assumptions to the conclusion.

What background knowledge is needed by a first-time learner of mathematical proof?

A high-school level understanding of algebra (integers, inequalities, basic functions) together with familiarity with the logical symbols ("and," "or," "implies," the universal quantifier ∀, and the existential quantifier ∃) is sufficient. A more rigorous knowledge of logic is developed from the intermediate section onward.

Why is it worthwhile to learn proof techniques?

Mastering proof techniques gives you a way to verify the correctness of mathematical statements and sharpens the logical thinking used throughout science, engineering, and computer science. The same skills are useful in practical settings such as verifying the correctness of algorithms or analyzing numerical error.