Welcome to the World of Mathematical Proof!
Ever wondered how mathematicians can be 100% sure that a rule works for every single number in existence? They don't just guess or test a few values; they use Mathematical Proof. In this chapter, you will learn the "rules of the game" for proving that mathematical statements are true (or false) once and for all.
Think of a proof like a legal case in court. You start with given assumptions, follow a series of logical steps, and reach a conclusion that no one can argue with. Let's dive in!
1. Proof by Deduction
Proof by deduction is the most common method you'll use in A Level Maths. It involves starting from known facts or "first principles" and using algebra to show that a statement must be true.
How it works:
Imagine a chain of dominoes. If the first one falls (your starting assumption), and every domino knocks over the next one (your logical steps), the last one is guaranteed to fall (your conclusion).
Step-by-Step Example:
Prove that \(n^2 - 6n + 10\) is positive for all values of \(n\).
1. Identify the tool: For quadratics, completing the square is our best friend because squares are always non-negative.
2. Complete the square: \(n^2 - 6n + 10 = (n - 3)^2 - 9 + 10\)
3. Simplify: \(= (n - 3)^2 + 1\)
4. State the logic: We know that any real number squared is at least zero, so \((n - 3)^2 \geq 0\).
5. Conclusion: Therefore, \((n - 3)^2 + 1 \geq 1\), which is always positive. Proof complete!
Quick Review Box:
● Always start with a general case (use letters like \(n\) or \(x\)), not specific numbers.
● Use completing the square to prove expressions are positive.
● Use differentiation from first principles (found later in the syllabus) as another form of deduction.
Key Takeaway: Deduction uses algebra to build a logical bridge from what you know to what you want to prove.
2. Proof by Exhaustion
Don't worry—this doesn't mean you have to work until you're tired! Proof by exhaustion means you break the problem down into every possible case and prove each one separately.
When to use it:
Use this when there are only a small, manageable number of possibilities to check.
Example from the Syllabus:
Given that \(p\) is a prime number such that \(3 < p < 25\), prove that \((p - 1)(p + 1)\) is a multiple of 12.
1. List the cases: The prime numbers between 3 and 25 are 5, 7, 11, 13, 17, 19, and 23.
2. Test Case 1 (\(p=5\)): \((5-1)(5+1) = 4 \times 6 = 24\). (Multiple of 12? Yes!)
3. Test Case 2 (\(p=7\)): \((7-1)(7+1) = 6 \times 8 = 48\). (Multiple of 12? Yes!)
4. Continue: You would repeat this for 11, 13, 17, 19, and 23. If all work, the proof is finished!
Did you know?
Computers are amazing at proof by exhaustion. In 1976, the "Four Color Theorem" was proved by a computer checking 1,936 different cases!
Key Takeaway: If you can't find a general rule, try splitting the problem into categories (like "even numbers" and "odd numbers") or checking a finite list of values.
3. Disproof by Counter-Example
This is often the most satisfying method. To disprove a statement (show it's false), you only need to find one single example where it doesn't work.
Analogy:
If someone says, "Every car in the world is blue," you don't need to look at every car. You just need to point at one red car to prove them wrong.
Example:
Disprove the statement: "\(n^2 - n + 1\) is a prime number for all values of \(n\)."
1. Try values: Let's test some numbers.
If \(n=1\), \(1^2 - 1 + 1 = 1\) (Not prime, but some definitions vary, so let's keep going).
If \(n=2\), \(2^2 - 2 + 1 = 3\) (Prime).
If \(n=3\), \(3^2 - 3 + 1 = 7\) (Prime).
2. Find the "oops": Let's try \(n=11\).
\(11^2 - 11 + 1 = 121 - 11 + 1 = 111\).
Wait! \(111\) is divisible by 3 (\(3 \times 37 = 111\)).
3. Conclusion: Since \(n=11\) produces a non-prime number, the statement is untrue.
Common Mistake to Avoid:
You cannot prove something is true by showing examples. You can only disprove it with examples. To prove something, you must use Deduction or Exhaustion.
Key Takeaway: One failure is enough to destroy a mathematical "rule."
4. Proof by Contradiction
This is the "secret agent" of math proofs. It feels a bit backwards at first, but it's very powerful. Don't worry if this seems tricky—it takes a bit of practice!
How it works:
1. Assume the statement you are trying to prove is FALSE.
2. Use logical steps to show this assumption leads to something impossible (a contradiction).
3. Therefore, your assumption must have been wrong, meaning the original statement must be TRUE.
Mandatory Proof: \(\sqrt{2}\) is Irrational
(Pearson Edexcel specifically requires you to know this proof!)
1. Assume the opposite: Assume \(\sqrt{2}\) is rational. This means it can be written as a fraction \(\frac{a}{b}\) in its simplest form (no common factors).
2. Square both sides: \(2 = \frac{a^2}{b^2}\), which means \(a^2 = 2b^2\).
3. The logic: This tells us \(a^2\) is even, so \(a\) must be even. Let's call \(a = 2k\).
4. Substitute: \((2k)^2 = 2b^2\) becomes \(4k^2 = 2b^2\), which simplifies to \(b^2 = 2k^2\).
5. The contradiction: This tells us \(b^2\) is even, so \(b\) must be even. But wait! If \(a\) and \(b\) are both even, the fraction wasn't in its simplest form. Contradiction!
6. Conclusion: Our assumption was wrong. \(\sqrt{2}\) must be irrational.
Memory Aid:
Think of Contradiction as "The Grumpy Method."
"Oh yeah? Let's assume you're right... [Math happens]... See? That makes no sense! You must be wrong."
Key Takeaway: If assuming the "opposite" leads to a mathematical disaster, then the original statement must be correct.
Summary Checklist for Your Exam
● Deduction: Can I use algebra (like completing the square) to show the result?
● Exhaustion: Are there just a few cases I can test one by one?
● Counter-Example: Can I find just one number that breaks the rule?
● Contradiction: Can I assume it's false and show that it leads to an impossibility? (Remember to memorize the proofs for \(\sqrt{2}\) and the infinity of primes!)