Welcome to the World of Mathematical Proof!
In your math journey so far, you have solved hundreds of equations and drawn dozens of graphs. But have you ever stopped to ask, "How do we know these rules actually work for every single number?"
That is where Proof comes in! Proof is the heart of Mathematics. It is the process of using logic to show that a statement is 100% true, 100% of the time. In this chapter of Pure Mathematics 2 (P2), we are going to learn how to build a solid argument and how to break a weak one.
Don't worry if this seems a bit "wordy" compared to other math topics. Once you see the patterns, it becomes a very satisfying puzzle to solve!
1. The Structure of a Mathematical Proof
A mathematical proof is like a recipe or a legal argument. You can't just jump to the end; you have to follow specific, logical steps to get there. Every proof generally follows this flow:
1. Assumptions: You start with things you already know are true (given facts or definitions).
2. Logical Steps: You use algebra and logic to move from one step to the next.
3. Conclusion: You arrive at the statement you were trying to prove.
Key Terms to Know
• Statement: A mathematical sentence that is either true or false (e.g., "The sum of two even numbers is even").
• Conjecture: A mathematical statement that we think is true but hasn't been proven yet.
• Theorem: A statement that has been proven to be true.
Quick Review: Prerequisite Basics
Before we start proving things, let's remember two definitions we use a lot:
• An Even Number can be written as \( 2n \), where \( n \) is an integer.
• An Odd Number can be written as \( 2n + 1 \), where \( n \) is an integer.
Takeaway: A proof is just a logical bridge that connects "what we know" to "what we want to show."
2. Proof by Exhaustion
This sounds tiring, doesn't it? But Proof by Exhaustion actually just means "checking every single possibility."
We use this method when there are only a limited number of cases to check. If you check every single case and the statement is true for all of them, then the whole statement is proven.
Analogy: The Hallway of Doors
Imagine you are in a hallway with only five doors. If you want to prove that "every door in this hallway is locked," you simply walk to each of the five doors and try to open them. Once you've checked all five, your proof is complete!
Example: Syllabus Challenge
Prove that if \( x \) and \( y \) are odd integers where \( x < 7 \) and \( y < 7 \), their sum is divisible by 2.
Step 1: List all possibilities.
The odd integers less than 7 are 1, 3, and 5.
Step 2: Test every combination of \( x \) and \( y \).
• \( 1 + 1 = 2 \) (Divisible by 2)
• \( 1 + 3 = 4 \) (Divisible by 2)
• \( 1 + 5 = 6 \) (Divisible by 2)
• \( 3 + 3 = 6 \) (Divisible by 2)
• \( 3 + 5 = 8 \) (Divisible by 2)
• \( 5 + 5 = 10 \) (Divisible by 2)
Step 3: Conclude.
Since we have checked every possible pair and the sum was always even, the statement is proven true by exhaustion.
Common Mistake to Avoid
The most common mistake is forgetting a case. If there are 6 possible combinations and you only check 5, your proof is not valid! Always make a list first to ensure you don't miss any.
Takeaway: Use exhaustion when the number of possibilities is small enough to list and check one by one.
3. Disproof by Counter-example
In Mathematics, for a rule to be "true," it must be true for every single case. This makes it very easy to prove a statement is false.
To disprove a statement, you only need to find one single example where the rule doesn't work. This is called a counter-example.
"Did you know?"
For centuries, people believed "All swans are white." This was a mathematical-style "truth" until someone discovered a black swan in Australia. That one black swan was a counter-example that disproved the "All swans are white" rule forever!
Example: The Prime Number Test
Disprove the statement: "\( n^2 - n + 1 \) is a prime number for all positive integers \( n \)."
Step 1: Try small values of \( n \).
• If \( n = 1 \): \( 1^2 - 1 + 1 = 1 \). (Wait, 1 is not a prime number!)
• If \( n = 2 \): \( 2^2 - 2 + 1 = 3 \). (Prime)
• If \( n = 3 \): \( 3^2 - 3 + 1 = 7 \). (Prime)
• If \( n = 5 \): \( 5^2 - 5 + 1 = 21 \). (Not prime! \( 3 \times 7 = 21 \))
Step 2: Present your counter-example.
"When \( n = 5 \), \( n^2 - n + 1 = 21 \). Since 21 is not a prime number, the statement is false."
Important Tip: Keep it Simple!
Don't worry if you find a very large counter-example, but usually, the examiners design questions so that a small number (like 0, 1, 2, or 5) will work. Always start testing with the easiest numbers first!
Takeaway: You don't need to show why a statement is usually wrong; you just need to show one instance where it isn't right.
Final Summary for Unit P2 Proof
• Mathematical Proof uses logical steps and clear definitions (like \( 2n \) for even numbers).
• Proof by Exhaustion means testing every case. It only works when the number of cases is small.
• Disproof by Counter-example is the fastest way to kill a false claim. Find one case where it fails, and the statement is officially disproved.
• Always be clear in your conclusion. State exactly what you have shown!
Keep practicing! Proof is a skill that gets much easier the more you "speak" the language of logic. You've got this!