Welcome to the World of Mathematical Proof!
Ever wondered how mathematicians can be 100% sure that something is true for every single number in existence? They don't just guess; they use Proof. In this chapter, you’ll learn how to build a rock-solid logical argument. Think of it like being a lawyer in a courtroom: you start with the evidence (facts we already know) and use logic to reach a verdict (the conclusion).
Don't worry if this seems a bit abstract at first! Once you learn the "recipes" for different types of proof, you'll find it’s one of the most rewarding parts of Pure Mathematics.
1. What is a Mathematical Proof?
A mathematical proof is a formal argument that shows a statement is always true. It follows a specific structure:
1. Assumptions: Starting with facts or definitions we know are true.
2. Logical Steps: A chain of reasoning where each step follows clearly from the previous one.
3. Conclusion: The final statement that you set out to prove.
Quick Review: Key Terms
- Integer: A whole number (like -2, 0, 5).
- Even Number: Can be written as \(2n\) (where \(n\) is an integer).
- Odd Number: Can be written as \(2n + 1\) (where \(n\) is an integer).
Key Takeaway: A proof isn't just showing that something works for a few numbers; it's showing it works for all possible cases within the rules.
2. Proof by Deduction
Proof by Deduction is the most common method. You start from known facts and use algebra to "deduce" the conclusion. It’s like a detective following a trail of clues to find the culprit.
Example: Using Completion of the Square
Suppose you are asked to prove that \(n^2 - 6n + 10\) is positive for all values of \(n\).
Step 1: Complete the square.
We rewrite the expression: \(n^2 - 6n + 10 = (n - 3)^2 - 9 + 10\)
Which simplifies to: \((n - 3)^2 + 1\)
Step 2: Use logic.
We know that any real number squared is always greater than or equal to zero.
So, \((n - 3)^2 \geq 0\).
Step 3: Reach the conclusion.
If we add 1 to a number that is at least zero, the result must be at least 1.
Therefore, \((n - 3)^2 + 1 \geq 1\), which means it is always positive.
Conclusion: \(n^2 - 6n + 10 > 0\) for all \(n\).
Common Mistake: Many students just try plugging in numbers like \(n=1\) or \(n=2\). While this shows it works for those numbers, it doesn't prove it for every number. You must use algebra (like completion of the square) to cover everything!
Key Takeaway: Use algebra to turn a general statement into an undeniable fact.
3. Proof by Exhaustion
This method sounds tiring, and that's because it involves testing every single possibility! You can only use this when there are a small, finite number of cases to check.
Analogy: If you want to prove that every light switch in your house works, "Proof by Exhaustion" means you walk into every room and flip every single switch yourself.
Example: Sum of Odd Integers
Statement: Prove that if \(x\) and \(y\) are odd integers less than 7 (and greater than 0), 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 all pairs.
- \(1 + 1 = 2\) (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: Conclusion.
Since we checked every possible pair and they all resulted in an even number, the statement is proven.
Did you know? Proof by exhaustion was used by computers to solve the famous "Four Color Map Theorem." There were nearly 2,000 cases to check—too many for a human, but easy for a computer!
Key Takeaway: If the group of numbers is small, just test them all!
4. Disproof by Counter-Example
Sometimes you are asked to show that a statement is false. To do this, you only need to find one single example where the statement doesn't work. This is called a counter-example.
Analogy: If someone claims "All cars are red," you don't need to look at every car in the world to prove them wrong. You just need to point at one blue car.
Example: Prime Numbers
Statement: "The expression \(n^2 - n + 1\) is a prime number for all values of \(n\)."
Step 1: Try small values of \(n\).
- If \(n = 1\): \(1^2 - 1 + 1 = 1\) (Note: 1 is not actually prime, so this is already a counter-example!)
- If \(n = 2\): \(2^2 - 2 + 1 = 3\) (Prime)
- If \(n = 3\): \(3^2 - 3 + 1 = 7\) (Prime)
- If \(n = 5\): \(5^2 - 5 + 1 = 21\)
Step 2: Identify the failure.
Wait! \(21\) is not a prime number because \(3 \times 7 = 21\).
Step 3: Conclusion.
Since the statement fails when \(n = 5\), the statement "is a prime number for all values of \(n\)" is untrue.
Quick Review: Common Counter-examples
When looking for a counter-example, always try these "sneaky" numbers first:
- 0
- 1
- Negative numbers (e.g., -1)
- Fractions (between 0 and 1)
Key Takeaway: One "failure" is enough to destroy a general mathematical claim.
Summary Checklist
Before you finish this chapter, make sure you can:
- [ ] Use Deduction (Algebra) to prove things like "even + even = even."
- [ ] Use Completion of the Square to prove an expression is always positive.
- [ ] Use Exhaustion to check a limited list of numbers.
- [ ] Find a Counter-example to show a statement is false.
Remember: Proof is about being precise. Always state your conclusion clearly at the end!