Welcome to the World of Mathematical Proof!
Hey there! Ready to become a mathematical detective? Proof is one of the most important parts of your AQA A-Level Maths journey. While most of maths is about *calculating* an answer, proof is about *proving* why a rule is true for every single number in existence.
Think of a proof like a court case: you can't just say someone is guilty; you need a solid, unbreakable chain of evidence. In this chapter, we will learn how to build that chain so that no one can find a hole in your logic.
1. The Basics: Speaking the Language
Before we start building proofs, we need to know the "building blocks." In maths, we use specific letters and terms to represent different types of numbers.
Key Terms to Know:
• Integers: Whole numbers (..., -2, -1, 0, 1, 2, ...). We usually use \(n\) or \(m\) to represent an integer.
• Even Numbers: Any number that can be written as \(2n\) (where \(n\) is an integer).
• Odd Numbers: Any number that is one more (or one less) than an even number, written as \(2n + 1\) or \(2n - 1\).
• Consecutive Integers: Numbers that follow each other, like \(n, n+1, n+2\).
• Rational Numbers: Numbers that can be written as a fraction \(\frac{p}{q}\).
Quick Review: If you see \(2n\), think "Even." If you see \(2n+1\), think "Odd." This simple trick is the "skeleton" of many proofs!
2. Proof by Deduction
Proof by deduction is the most common method. You start from known facts (like the definitions above) and use logical steps to reach a conclusion.
Step-by-Step Example:
Prove that the sum of any two even numbers is always even.
Step 1: Define your numbers.
Let the first even number be \(2n\) and the second even number be \(2m\), where \(n\) and \(m\) are integers.
Step 2: Do the math.
Add them together: \(2n + 2m\).
Step 3: Factorise to show the pattern.
\(2n + 2m = 2(n + m)\).
Step 4: Conclude.
Since \((n + m)\) is an integer, \(2(n + m)\) fits the definition of an even number. Proof complete!
Common Mistake to Avoid: Don't use the same letter for both numbers (like \(2n + 2n\)) unless the question specifically says the numbers are the same. Using \(n\) and \(m\) keeps it general!
Key Takeaway: Deduction is about using algebra to show that a statement *must* be true based on definitions.
3. Proof by Exhaustion
This method sounds tiring, and sometimes it is! Proof by exhaustion means breaking the problem down into every possible case and proving each one separately.
Analogy: Imagine you want to prove all the lightbulbs in your house work. Proof by exhaustion is walking into every single room and flicking every switch.
Example:
Prove that \(n^2 + n\) is even for all positive integers \(n \leq 3\).
Case 1 (\(n=1\)): \(1^2 + 1 = 2\) (Even)
Case 2 (\(n=2\)): \(2^2 + 2 = 6\) (Even)
Case 3 (\(n=3\)): \(3^2 + 3 = 12\) (Even)
All cases are even, so the statement is proved by exhaustion.
When to use it? Use this when there are only a few possible values to check, or when you can split all numbers into two groups (like "Even numbers" and "Odd numbers").
4. Disproof by Counter-example
This is the "easiest" type of proof. To disprove a statement, you only need to find one single case where it doesn't work.
Did you know? In maths, "always" means 100% of the time. If it fails once, the whole statement is officially "False."
Example:
Disprove the statement: "All prime numbers are odd."
Counter-example: Consider the number 2. 2 is a prime number, but it is even. Therefore, the statement is false.
Key Takeaway: You don't need a fancy formula for disproof. Just find one "rebel" number that breaks the rule!
5. Proof by Contradiction
This is a clever "reverse psychology" method. Don't worry if this feels tricky at first; it's a bit of a brain-bender!
How it works:
1. Assume the statement is FALSE.
2. Use logic until you hit a "mathematical crash" (a contradiction).
3. Since your logic was perfect, the only mistake must have been your starting assumption.
4. Therefore, the original statement must be TRUE.
The Famous Example: \(\sqrt{2}\) is Irrational
1. Assume the opposite: Assume \(\sqrt{2}\) is rational. This means \(\sqrt{2} = \frac{p}{q}\) where the fraction is in its simplest form (no common factors).
2. Square both sides: \(2 = \frac{p^2}{q^2}\) which means \(p^2 = 2q^2\).
3. Logic: This means \(p^2\) is even, so \(p\) must be even. Let \(p = 2k\).
4. Substitute: \((2k)^2 = 2q^2 \implies 4k^2 = 2q^2 \implies 2k^2 = q^2\).
5. The Crash: This means \(q^2\) is also even, so \(q\) must be even.
6. The Contradiction: If \(p\) and \(q\) are both even, the fraction \(\frac{p}{q}\) wasn't in its simplest form! This contradicts our first step.
7. Conclusion: Our assumption was wrong. \(\sqrt{2}\) must be irrational.
Key Takeaway: Contradiction is great for proving things *don't* exist or are *not* something (like "not rational" or "not finite").
Quick Tips for Exam Success
• Read carefully: Does it say "for all integers" or "for all positive integers"? Zero and negative numbers can be great counter-examples!
• State the obvious: Always write a concluding sentence like "Since this is a multiple of 2, the expression is even."
• Don't panic: If you're stuck on a proof, try a few numbers first. It might help you see the pattern for a deductive proof or find a counter-example.
Final Summary Checklist:
• Deduction: Use algebra to move from start to finish.
• Exhaustion: Check every single case or group.
• Counter-example: Find one "fail" to kill a theory.
• Contradiction: Assume the opposite and find a logic error.