Disproof by Counter Example
Learn to disprove false statements using a single example
SPEC 1.1 • PROOF
When to Use Counter Examples
Disproof by counter example is used to show a statement is FALSE by finding just ONE example where the statement fails.
Statement TypeExampleWhat to Do
"All primes are odd"Universal claimFind ONE even prime
"n² > n for all n"Inequality claimFind ONE n where n² ≤ n
"2ⁿ − 1 is always prime"Always true claimFind ONE n where 2ⁿ − 1 is not prime
Important
Counter examples can only disprove statements. You cannot use examples to prove something is always true!
Finding Counter Examples
To find a counter example efficiently, try these special values first:
Try These First
n = 0 — often breaks statements
n = 1 — boundary case
n = 2 — smallest prime, only even prime
Negative numbers — if allowed
Fractions — if domain is reals
Why These Work
0 and 1 are edge cases with special properties
2 is the only even prime
Negatives flip inequalities
Fractions between 0 and 1 behave differently
Strategy
Read the statement carefully. Think: "What type of number might break this?" Then test systematically.
Writing Your Answer
When presenting a counter example in an exam, follow this structure:
Answer Structure
State your counter example: "Consider n = ..." or "Let x = ..."
Show it satisfies the conditions: Verify any requirements
Show the statement fails: Calculate and show the claim is false
Conclude: "This is a counter example, so the statement is false."
Exam Tip
Always write a clear conclusion. Don't just show the calculation - explicitly state that you've found a counter example and the statement is therefore false.
Worked Example01
Disprove the statement: "All prime numbers are odd."
1
We need to find ONE prime number that is NOT odd (i.e., is even).
2
Consider n = 2.
3
2 is prime (its only factors are 1 and itself).
4
2 is even (it is divisible by 2).
Conclusion: The number 2 is a counter example. It is prime but not odd. Therefore, the statement "all prime numbers are odd" is FALSE.
Worked Example02
Disprove the statement: "n² > n for all positive integers n."
1
We need to find ONE positive integer where n² is NOT greater than n.
2
Consider n = 1.
3
When n = 1: n² = 1² = 1 and n = 1.
4
So n² = 1 and n = 1, which means n² = n, not n² > n.
Conclusion: When n = 1, we have n² = n, not n² > n. Therefore, n = 1 is a counter example and the statement is FALSE.
Worked Example03
Disprove the statement: "2ⁿ − 1 is prime for all positive integers n > 1."
1
We need to find ONE value of n > 1 where 2ⁿ − 1 is NOT prime.
2
Let's test systematically:
n = 2: 2² − 1 = 3 ✓ (prime)
n = 3: 2³ − 1 = 7 ✓ (prime)
n = 4: 2⁴ − 1 = 15 = 3 × 5 ✗ (not prime!)
3
When n = 4: 2⁴ − 1 = 16 − 1 = 15 = 3 × 5.
Conclusion: When n = 4, we get 2⁴ − 1 = 15, which is not prime since 15 = 3 × 5. Therefore, n = 4 is a counter example and the statement is FALSE.
Worked Example04
Disprove the statement: "n² + n + 41 is prime for all positive integers n."
1
This formula famously gives primes for n = 1, 2, 3, ..., 39. But we need ONE counter example.
2
Try n = 41 (since 41 appears in the formula, this might cause divisibility).
3
When n = 41:
n² + n + 41 = 41² + 41 + 41 = 41² + 2(41) = 41(41 + 2) = 41 × 43
4
41 × 43 = 1763, which is not prime.
Conclusion: When n = 41, we get 41² + 41 + 41 = 41 × 43 = 1763, which is not prime. Therefore, n = 41 is a counter example and the statement is FALSE.
Worked Example05
Disprove the statement: "If n is a positive integer, then n³ > n²."
1
We need to find ONE positive integer where n³ is NOT greater than n².
2
Consider n = 1 (a boundary value).
3
When n = 1:
n³ = 1³ = 1
n² = 1² = 1
4
Since 1 = 1, we have n³ = n², not n³ > n².
Conclusion: When n = 1, we have 1³ = 1² = 1, so n³ is not greater than n². Therefore, n = 1 is a counter example and the statement is FALSE.
Worked Example06
Disprove the statement: "For all real numbers x, if x² > 4 then x > 2."
1
We need to find ONE real number where x² > 4 but x is NOT > 2.
2
Consider x = −3 (a negative number).
3
When x = −3:
x² = (−3)² = 9, and 9 > 4 ✓
4
But x = −3, and −3 is NOT greater than 2 ✗
Conclusion: When x = −3, we have x² = 9 > 4, but x = −3 < 2. Therefore, x = −3 is a counter example and the statement is FALSE.
Worked Example07
Disprove the statement: "The sum of two irrational numbers is always irrational."
1
We need to find TWO irrational numbers whose sum is rational.
2
Consider the irrational numbers √2 and −√2.
3
√2 is irrational (cannot be written as a fraction).
−√2 is also irrational (negative of an irrational is irrational).
4
Their sum: √2 + (−√2) = 0, which is rational.
Conclusion: √2 and −√2 are both irrational, but their sum is 0, which is rational. Therefore, this is a counter example and the statement is FALSE.
Practice
Question 1: Disprove the statement: "n² + 3n + 2 is prime for all positive integers n."
Counter example: n = 1
When n = 1: n² + 3n + 2 = 1 + 3 + 2 = 6
6 = 2 × 3, which is not prime.
Conclusion: When n = 1, we get 6, which is not prime. Therefore, n = 1 is a counter example and the statement is FALSE.
Note: We could also factor: n² + 3n + 2 = (n + 1)(n + 2), which is always composite for n ≥ 1.
Practice
Question 2: Disprove the statement: "If a and b are irrational, then ab is irrational."
Counter example: a = √2 and b = √2
√2 is irrational (given).
ab = √2 × √2 = 2
2 is rational (it can be written as 2/1).
Conclusion: Both √2 and √2 are irrational, but their product is 2, which is rational. Therefore, this is a counter example and the statement is FALSE.
Practice
Question 3: Disprove the statement: "For all integers n > 0, 2n + 1 is not divisible by 3."
Counter example: n = 1
When n = 1: 2n + 1 = 2(1) + 1 = 3
3 is divisible by 3 (3 ÷ 3 = 1).
Conclusion: When n = 1, we get 2n + 1 = 3, which IS divisible by 3. Therefore, n = 1 is a counter example and the statement is FALSE.
Note: n = 4 also works: 2(4) + 1 = 9 = 3 × 3.
Summary
Key Concepts
ConceptDescription
Counter ExampleA specific case where a statement fails
One is EnoughYou only need ONE example to disprove
Values to Try0, 1, 2, negatives, fractions
ConclusionAlways state the statement is FALSE
Key Exam Skills
Read carefully - if asked to disprove, a counter example exists
Try boundary values: 0, 1, 2, −1 first
Show full working: state counter example, verify conditions, show failure
Write clear conclusion stating the statement is false
Next: When to Use Counter Examples