Welcome to the World of Mathematical Proof!
Ever wondered how we know for certain that a mathematical rule works every single time, for every possible number? We don't just "guess" or "hope" it works; we use Proof. Think of a proof as a logical roadmap. It takes us from a starting point (something we know is true) to a finish line (a new conclusion) using unbreakable steps. In this chapter, you’ll learn how to be a mathematical "lawyer," building cases that are so solid they can never be argued against!
1. The Building Blocks of a Proof
Before we start "building," we need our tools. A mathematical proof usually follows a specific structure:
1. Assumptions: These are the starting facts we know are true (like "let \(n\) be an even integer").
2. Logical Steps: A chain of reasoning where each step follows naturally from the one before.
3. Conclusion: The final statement that you have successfully proven.
Quick Review: Key Terms
- Statement: A sentence that is either true or false (e.g., "7 is a prime number").
- Variable: A letter representing a number that can change (like \(x\) or \(n\)).
- Constant: A fixed number that doesn't change (like \(5\) or \(\pi\)).
- Identity \(\equiv\): A statement that is true for all values of the variable. It’s like a super-equals sign!
Key Takeaway
A proof is not just showing that something works for a few numbers; it is a logical argument that shows it works for every possible case.
2. Proof by Deduction
Proof by deduction is the most common method. You start from known facts and use algebra to "deduce" the result. It’s like following a recipe: if you start with the right ingredients and follow the steps, you always get the same cake.
Example: Prove that the sum of any two even numbers is always even.
Step-by-step Explanation:
1. Define your numbers: Any even number can be written as \(2n\), where \(n\) is an integer. Let our two even numbers be \(2m\) and \(2k\).
2. Add them together: \(2m + 2k\).
3. Factorise: We can pull out a 2, giving us \(2(m + k)\).
4. Conclude: Since \(m + k\) is just another integer, any number in the form \(2 \times (\text{integer})\) must be even. Proof complete!
Don't worry if this seems tricky at first! The "trick" is often just finding the right way to write the numbers. Here is a handy memory aid:
Memory Aid: How to write numbers in proofs
- Even numbers: \(2n\)
- Odd numbers: \(2n + 1\) (or \(2n - 1\))
- Consecutive numbers: \(n, n+1, n+2\)...
Key Takeaway
In deduction, you use algebra to turn your starting point into your conclusion. If you can show the final answer fits the "form" of an even or odd number, you've won!
3. Proof by Exhaustion
Sometimes, algebra is too complicated, but there are only a few possible cases to check. Proof by exhaustion means testing every single possibility until there are none left to check. It's called "exhaustion" because it can be quite tiring if there are many cases!
Analogy: Imagine you want to prove that every student in a small classroom of five people is wearing shoes. Instead of using a complex theory, you just look at student 1, then student 2, then 3, 4, and 5. Once you've checked everyone, you've proven it!
Example: Prove that \(n^2 + 2\) is not divisible by 4 for \(n = 1, 2, 3\).
- Case 1: If \(n = 1\), \(1^2 + 2 = 3\). (3 is not divisible by 4).
- Case 2: If \(n = 2\), \(2^2 + 2 = 6\). (6 is not divisible by 4).
- Case 3: If \(n = 3\), \(3^2 + 2 = 11\). (11 is not divisible by 4).
All cases have been checked, so the statement is proven.
Common Mistake: Students often forget a case. If the question asks you to prove something for "all integers," you usually cannot use exhaustion because there are infinitely many integers!
Key Takeaway
Use exhaustion only when there are a small, finite number of cases to check. Make sure you list and check every single one.
4. Disproof by Counter-Example
In mathematics, for a rule to be "true," it must be true 100% of the time. If you find even one case where the rule doesn't work, the whole rule is destroyed. This is called a counter-example.
Did you know? You don't need a complicated reason to disprove a statement. Finding just one single number that fails is enough to win the argument!
Example: Disprove the statement "All prime numbers are odd."
- The Counter-example: The number 2.
- Explanation: 2 is a prime number, but 2 is even. Therefore, the statement is false.
Step-by-step for finding a counter-example:
1. Look at the claim (e.g., "The square of any number is larger than the number itself").
2. Try "unusual" numbers like 0, 1, negatives, or fractions.
3. Test them: If \(n = 0.5\), then \(n^2 = 0.25\). Since \(0.25\) is NOT larger than \(0.5\), the statement is disproven!
Key Takeaway
To prove something is true, you need a full argument (Deduction or Exhaustion). To prove something is false, you only need one counter-example.
5. Critiquing Arguments
The syllabus also wants you to be able to "critique" or judge other people's proofs. When looking at a proof, ask yourself:
- Is the language precise? (Are they using coefficient, term, and expression correctly?)
- Does one step actually lead to the next?
- Did they use the correct symbols, like \(\equiv\) for identities?
Quick Review Box
- Deduction: Use algebra/logic to show it's always true.
- Exhaustion: Check every single case individually.
- Counter-example: Find one case that fails to disprove the whole thing.
Don't be discouraged if proofs feel different from the rest of maths. You are learning a new way of thinking! Keep practicing the "even/odd" algebraic setups, as those are very common in Paper 1.