How To Write A Mathematical Proof

How to Write a Mathematical Proof: A Comprehensive Guide

Mathematical proofs are the backbone of mathematical reasoning. They provide rigorous, logical arguments to establish the truth of mathematical statements, known as theorems. Learning to write a mathematical proof is a crucial skill for anyone pursuing studies in mathematics, computer science, or related fields. This comprehensive guide will equip you with the knowledge and strategies to craft clear, concise, and convincing mathematical proofs.

Understanding the Fundamentals

Before diving into the intricacies of proof writing, it's essential to grasp fundamental concepts:

1. Definitions and Axioms: The Building Blocks

Mathematical proofs rely on precisely defined terms and accepted axioms. Definitions provide the meaning of concepts, while axioms are self-evident truths or assumptions upon which the entire mathematical system is built. Understanding these building blocks is paramount to constructing a valid argument.

• Example: The definition of a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Axioms in Euclidean geometry include statements like "a straight line can be drawn between any two points."

2. Theorems and Conjectures: What Needs Proving

Theorems are statements that have been proven to be true. Conjectures, on the other hand, are statements that are believed to be true but haven't yet been proven. The goal of a proof is to transform a conjecture into a theorem.

• Example: The Pythagorean theorem (a² + b² = c²) is a theorem. The Goldbach conjecture (every even integer greater than 2 can be expressed as the sum of two primes) is a famous conjecture that remains unproven.

3. Logic and Deductive Reasoning: The Engine of Proof

Mathematical proofs employ deductive reasoning, moving from general principles to specific conclusions. Each step in the proof must logically follow from previous steps, definitions, axioms, or previously proven theorems. The core principles of logic, such as implication, conjunction, and disjunction, play a crucial role.

Common Proof Techniques

Several established techniques are frequently employed in mathematical proofs. Mastering these strategies is key to crafting effective arguments:

1. Direct Proof: The Straightforward Approach

A direct proof starts with the premises (given information) and, through a series of logical steps, directly arrives at the conclusion. This is often the most intuitive approach.

• Example: To prove that the sum of two even integers is even, we start with two arbitrary even integers, 2m and 2n, where m and n are integers. Their sum is 2m + 2n = 2(m + n). Since m + n is an integer, the sum is a multiple of 2, hence even.

2. Proof by Contradiction (Reductio ad Absurdum): The Indirect Approach

This powerful technique assumes the negation of the statement to be proven and then demonstrates that this assumption leads to a contradiction. Since a contradiction cannot exist within a consistent mathematical system, the original statement must be true.

• Example: To prove that √2 is irrational, we assume it's rational (√2 = a/b, where a and b are coprime integers). Manipulating this equation leads to a contradiction, demonstrating that our assumption was false, thus proving √2 is irrational.

3. Proof by Induction: Climbing the Ladder

Mathematical induction is used to prove statements about all natural numbers. It involves two steps:

• Base case: Prove the statement holds for the smallest natural number (usually 1).

• Inductive step: Assume the statement holds for some arbitrary natural number k (inductive hypothesis) and then prove it holds for k+1. This shows that if the statement holds for one number, it holds for the next, creating a chain reaction that proves it for all natural numbers.

• Example: To prove that the sum of the first n natural numbers is n(n+1)/2, the base case (n=1) is trivial. The inductive step involves assuming the formula holds for k and then proving it holds for k+1 by showing that the sum of the first k+1 numbers is (k+1)(k+2)/2.

4. Proof by Cases: Covering All Possibilities

When a statement applies differently under various conditions, a proof by cases can be used. The statement is proven for each case separately, covering all possibilities.

• Example: To prove that |x| ≥ 0 for all real numbers x, we consider two cases: x ≥ 0 and x < 0. In each case, the inequality holds true.

5. Proof by Contrapositive: Reversing the Implication

The contrapositive of a statement "If P, then Q" is "If not Q, then not P". A statement and its contrapositive are logically equivalent, so proving the contrapositive proves the original statement.

• Example: To prove "If n² is even, then n is even," it's easier to prove its contrapositive: "If n is odd, then n² is odd."

Writing Style and Structure: Clarity and Precision

The elegance and effectiveness of a mathematical proof extend beyond its logical validity. Clear writing and well-structured arguments are crucial for readability and comprehension.

1. Begin with a Clear Statement of the Theorem

Explicitly state the theorem you intend to prove at the beginning. This provides context and makes the goal clear.

2. Define Symbols and Variables

Clearly define all symbols and variables used in the proof. Ambiguity can lead to misunderstandings.

3. Number Your Steps

Numbering steps aids in tracking the logical progression and facilitates referencing specific parts of the proof.

4. Explain Each Step

Don't just present equations; explain the reasoning behind each step. This shows your thought process and allows readers to follow your argument.

5. Use Precise Language

Avoid vague terms and use precise mathematical language. Words like "therefore," "hence," "thus," and "consequently" signal logical connections.

6. Conclude with a Summary

Summarize the results and explicitly state that the theorem has been proven.

Advanced Proof Techniques

As you progress in your mathematical studies, you may encounter more sophisticated proof techniques. These include:

• Pigeonhole principle: If you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. This simple principle has surprising applications in advanced proofs.
• Counting arguments: Using combinatorial techniques to demonstrate the existence or properties of mathematical objects.
• Diagonalization arguments: Used to show the uncountability of certain sets, such as the real numbers.
• Epsilon-delta proofs: Fundamental in analysis, these proofs rigorously define limits and continuity.

Practice Makes Perfect

Writing mathematical proofs is a skill that develops with practice. Start with simple proofs and gradually work your way towards more complex ones. Seek feedback on your proofs from instructors or peers, and don't be afraid to revise and refine your work.

Conclusion

Mastering the art of writing mathematical proofs is a rewarding journey. By understanding fundamental concepts, employing appropriate techniques, and developing a clear writing style, you can construct rigorous, elegant, and convincing arguments. Consistent practice and a willingness to learn from feedback are key to success in this crucial aspect of mathematical reasoning. Remember that a well-written proof is not only logically correct but also readily understandable and appreciated by others. The clarity and precision of your arguments reflect your understanding and mastery of the subject matter.