how to write a proof math

Guri Bee

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17-10-2024

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Demystifying Mathematical Proofs: A Step-by-Step Guide

Mathematical proofs are the backbone of mathematical understanding. They provide rigorous justifications for mathematical statements, ensuring their validity and forming the foundation for further exploration. But tackling a proof can feel daunting for beginners. This article aims to guide you through the process, demystifying the art of writing proofs.

What is a mathematical proof?

A proof is a logical argument that establishes the truth of a mathematical statement, called a theorem. It starts with known facts (axioms, definitions, or previously proven theorems) and uses logical reasoning to arrive at the conclusion.

Essential Elements of a Proof

1. Clear Statement of the Theorem: Begin by clearly stating the theorem you intend to prove.
2. Assumptions: Explicitly state any assumptions or given information.
3. Logical Steps: Present a sequence of logically sound steps that connect the assumptions to the conclusion.
4. Definitions and Properties: Use precise definitions and relevant mathematical properties in your argument.
5. Justification: Provide reasons for each step, citing definitions, axioms, or previously proven theorems.
6. Conclusion: Clearly state the conclusion, indicating that you have successfully proven the theorem.

Common Proof Techniques

1. Direct Proof: Start with the assumptions and use logical deduction to arrive at the conclusion.

   • Example: Prove that the sum of two even numbers is always even.
   • Proof: Let 'a' and 'b' be two even numbers. By definition, 'a' can be written as 2m and 'b' as 2n, where m and n are integers. Their sum, a + b, is equal to 2m + 2n, which can be factored as 2(m + n). Since m + n is also an integer, a + b is a multiple of 2, making it an even number.

2. Proof by Contradiction: Assume the opposite of what you want to prove and show that this leads to a contradiction.

   • Example: Prove that the square root of 2 is irrational.
   • Proof: Assume the opposite, that the square root of 2 is rational. This means it can be expressed as a fraction p/q, where p and q are integers with no common factors. Squaring both sides gives 2 = p^2 / q^2, implying 2q^2 = p^2. This means p^2 is even, which in turn implies that p itself is even. Therefore, p can be written as 2k, where k is an integer. Substituting this into the equation 2q^2 = p^2 gives 2q^2 = (2k)^2, which simplifies to q^2 = 2k^2. This shows that q^2 is also even, and therefore q is even. But this contradicts our initial assumption that p and q have no common factors. Since our assumption leads to a contradiction, it must be false, proving that the square root of 2 is irrational.

3. Proof by Induction: Prove a statement for a base case and then show that if the statement is true for a particular case, it also holds for the next case.

   • Example: Prove that the sum of the first n natural numbers is n(n+1)/2.
   • Proof:
     • Base Case: For n=1, the sum is 1, which is equal to 1(1+1)/2.
     • Induction Step: Assume the statement is true for some k, i.e., the sum of the first k natural numbers is k(k+1)/2. We need to show that the statement is also true for k+1. The sum of the first k+1 natural numbers is k(k+1)/2 + (k+1), which simplifies to (k+1)(k+2)/2. This proves the statement for k+1.

Important Tips

• Practice, Practice, Practice: The key to mastering proofs is consistent practice. Work through problems and study different proof techniques.
• Seek Clarity: Ensure that every step in your proof is logically connected and easily understandable.
• Seek Feedback: Ask a friend or tutor to review your proofs and provide feedback.

Conclusion

Mathematical proofs are a powerful tool for establishing truth and building a solid foundation for further mathematical exploration. While they may seem complex at first, with practice and understanding, you can unlock the beauty and logic behind these elegant arguments.