Disproof by Counter Example (Edexcel A-Level Mathematics): Revision Notes

Step 1: Understand the Statement

First, make sure you understand the statement you're trying to disprove. Identify whether it is a general statement (often using words like "all", "every", or "for every") that claims something is true for a broad set of values.

For example, consider the statement: "All numbers that are multiples of 2 are also multiples of 4."

Step 2: Identify What You Need to Disprove

You are looking for one example that does not satisfy the statement. In this case, you're trying to find a number that is a multiple of 2 but not a multiple of 4.

Step 3: Test Examples

Now, start testing specific examples that meet the first part of the statement. For our example, choose some numbers that are multiples of 2:

• 2, 4, 6, 8, 10...

Step 4: Find a Counterexample

Look for an example that breaks the statement. In our case, 6 is a multiple of 2, but it is not a multiple of 4 (since 6 ÷ 4 does not give a whole number).

Therefore, 6 is the counterexample that disproves the original statement.

Step 5: Conclude

Once you've found a counterexample, you can conclude that the statement is false. You don't need to check all cases—just one counterexample is enough to disprove the statement.

Summary

1. Understand the statement you're trying to disprove.
2. Identify what kind of example would contradict the statement.
3. Test specific examples that fit part of the statement.
4. Find a counterexample that breaks the statement.
5. Conclude that the statement is false.

Example: Disprove the following statement by mean of $n$ counterexample

• Statement: $2^n - 1$ is always prime for any positive integer $n \geq 2$.
• Counterexample:
• $n = 4$
• $2^4 - 1 = 16 - 1 = 15$
• $15 = 5 \times 3$ (not prime)
• Important: Finding a counterexample is a very important step.

Example Problem: All prime numbers are odd.

1. Identify a counterexample:

• Consider the prime number $2$.

2. Check if it contradicts the statement:

• The number $2$ is a prime number because its only divisors are $1$ and $2$.
• However, $2$ is not odd; it is even.

3. Conclusion:

• The existence of the prime number $2$, which is even, contradicts the statement that all prime numbers are odd. Therefore, the statement "All prime numbers are odd" is false. The counterexample ($2$) successfully disproves it.