Proof — Introduction
Foundations of propositions and logic
What you will learn in this section
This is the first step toward understanding what a mathematical "proof" is. No difficult notation is used; the exposition begins with everyday language.
- What is a "proposition"
- The meaning of "and," "or," and "if … then"
- How to read a simple proof
Contents
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Chapter 1
What is a proposition?
Sentences whose truth or falsity is determined; examples and non-examples.
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Chapter 2
Logical operations (and / or / not)
The meaning of AND, OR, NOT and their truth tables.
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Chapter 3
The meaning of "if … then"
A correct understanding of the conditional P → Q.
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Chapter 4
Converse, inverse, contrapositive
Transformations of a conditional and equivalence relations.
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Chapter 5
Reading proofs
Understanding the structure of simple proofs.
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Chapter 6
Induction and deduction
The difference between deduction and induction, and why mathematical induction is deductive.
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Chapter 7
Exercises
A summary of the introductory section with practice problems (with example solutions).
Prerequisites
No special prerequisites are required. Material at the level of junior-high-school mathematics — such as properties of integers, basic algebraic manipulations, and the use of variables — is used as examples.
Frequently Asked Questions
What does the introduction to proof cover?
The basic concept of "what a proof is," the distinction between propositions, axioms, and theorems, the basics of logical symbols (and, or, if … then, not), and the conventional formats for writing proofs. This is a starting point aimed at learners who are studying mathematical proof for the first time.
How does a mathematical proof differ from an explanation?
An "explanation" aims to convince through intuition or examples, whereas a "proof" is a process that starts from assumptions and derives the conclusion using logical rules alone — even a single counterexample makes the proof invalid. A proof is a reproducible logical procedure that allows any reader to reach the same conclusion.
How should one practice in order to learn to write proofs?
First read existing proofs carefully and analyze "which assumptions were used" and "which theorem was applied at each step." Then practice writing your own proofs of simple propositions (sums of even numbers, products of consecutive integers, etc.). Begin by imitating formal phrasing and gradually move toward expressing the argument in your own words.