Disproof by counterexample

Welcome to the World of Mathematical Detective Work!

In your H2 Mathematics journey, you spent a lot of time proving that things are true. But in H3 Mathematics (9820), we also care about proving when things are false. Imagine someone makes a bold claim like, "Every person in Singapore loves spicy food." To prove them wrong, you don’t need to survey the entire population. You just need to find one person who hates chili. That single person is your counterexample.

In this chapter, we will learn how to use this powerful tool to dismantle incorrect mathematical conjectures. Don't worry if this seems a bit "backwards" at first—it’s actually one of the most satisfying parts of mathematical reasoning!

1. What Exactly is a Counterexample?

A counterexample is a specific case or example that shows a general statement is false. In the language of logic, we use counterexamples to disprove universal statements (statements that claim something is true "for all" cases).

The Logic Behind It

Suppose we have a statement: "For all \(x\), if \(x\) satisfies property \(P\), then \(x\) satisfies property \(Q\)."
To disprove this, we only need to find one specific value of \(x\) such that:
1. \(x\) does satisfy property \(P\) (the condition is met).
2. \(x\) does not satisfy property \(Q\) (the conclusion fails).

Quick Tip: You only need one working counterexample to kill a theory. It doesn't matter if the statement is true for a billion other cases; if it fails once, the general statement is officially false.

Key Takeaway: A counterexample is the "exception to the rule" that proves the rule isn't actually a rule!

2. When Do We Use Disproof by Counterexample?

We use this method when we are given a conjecture (a mathematical guess) that involves the "For all" (\(\forall\)) quantifier.

Did you know? Many famous mathematicians spent years trying to prove conjectures, only for someone else to find a single counterexample that rendered their work invalid. It's the ultimate "Gotcha!" moment in math.

Common "Quantifiers" to Watch For:

• "For all \(n \in \mathbb{Z}^+\)..."
• "For every real number \(x\)..."
• "Given any prime number \(p\)..."

If you see these phrases and suspect the statement might be wrong, start hunting for a counterexample!

3. How to Find a Counterexample: A Step-by-Step Guide

Finding a counterexample is like being a detective. You need to look in the "likely places" where a rule might break down. Here is a process you can follow:

Step 1: Understand the Claim. Identify exactly what the "If" (antecedent) and "Then" (consequent) parts are.
Step 2: Test "Normal" Cases. Try small numbers like 2, 3, or 5. If the statement holds, it might actually be true (and you'd need a different proof method).
Step 3: Test "Boundary" or "Special" Cases. This is where most counterexamples hide! Try:
• The number 0 or 1.
• Negative numbers.
• Fractions between 0 and 1.
• Very large numbers.
• Prime numbers vs. Composite numbers.
• Even numbers vs. Odd numbers.
Step 4: Verify. Once you find a candidate, double-check that it meets the initial conditions but fails the conclusion.

4. Real-World and Mathematical Examples

Example 1: Algebra

Conjecture: "For all real numbers \(a\) and \(b\), \((a + b)^2 = a^2 + b^2\)."
Search: Let's test some values. If \(a = 1\) and \(b = 1\)...
Left Hand Side (LHS): \((1 + 1)^2 = 2^2 = 4\)
Right Hand Side (RHS): \(1^2 + 1^2 = 1 + 1 = 2\)
Conclusion: Since \(4 \neq 2\), the case \(a = 1, b = 1\) is a counterexample. The statement is disproved.

Example 2: Number Theory

Conjecture: "For every positive integer \(n\), \(n^2 + n + 41\) is a prime number."
Search: If you test \(n = 1, 2, 3...\), it actually looks true! (This is a famous "trick" conjecture).
But let's look for a special case. What if we make the expression easy to factorize? Let's try \(n = 41\).
When \(n = 41\), the expression becomes \(41^2 + 41 + 41\).
We can factor out 41: \(41(41 + 1 + 1) = 41 \times 43\).
Conclusion: Since \(41 \times 43\) is not prime, \(n = 41\) is a counterexample. The statement is false.

Memory Aid: Think of "The 4 O's" when searching for counterexamples: One, Other side (negatives), Oddities (like 0), and Outliers (extreme values).

5. Common Mistakes to Avoid

Even the best H3 students can trip up here. Watch out for these pitfalls:

• Picking a value that violates the premise: If the question says "For all prime numbers," you cannot use \(n = 4\) as a counterexample because 4 isn't prime to begin with.
• Arguing with words instead of examples: Don't just say "It's not always true because some numbers are different." You must provide a specific, numerical example.
• Assuming one counterexample isn't enough: You don't need to show it's "usually" false. One case is total destruction of the claim!

Quick Review Box:
1. Target: Statements claiming something is always true (\(\forall\)).
2. Goal: Find ONE case where the "If" is true but the "Then" is false.
3. Success: The statement is now considered "Disproved."

6. Connecting to Other Proofs

In your H3 syllabus, you will also study Proof by Contradiction. While they sound similar, they are different tools:

• Counterexample: Used to show a specific "For all" statement is false.
• Contradiction: Used to show a statement is true by showing that its negation leads to an impossible situation.

Summary Takeaway: Disproof by counterexample is the most efficient way to reject a false conjecture. It requires creativity, a bit of "rule-breaking" spirit, and a solid understanding of mathematical properties.

Don't worry if you can't find a counterexample immediately. Sometimes they are well-hidden! Keep testing different types of numbers, and eventually, you'll crack the case.