1.1.2 Disproof by counter example

What you'll learn

What conjectures and counterexamples are.
Why one counterexample can disprove a universal claim.
How to search efficiently for a useful counterexample.
How to write a complete, logically sound disproof.

Conjectures and proof

A conjecture is a mathematical statement that is believed to be true but has not yet been proved. It may arise from noticing a pattern, testing examples or considering a mathematical model.

For example, after calculating

12+1+41=43,22+2+41=47,32+3+41=53,1^2+1+41=43,\qquad 2^2+2+41=47,\qquad 3^2+3+41=53,

you might conjecture that $n^2+n+41$ is prime for every positive integer $n$ .

Checking several examples can provide evidence for a conjecture, but evidence is not the same as proof. A proof must show that the statement is true in every case covered by the conjecture.

Definition

Universal statement

A universal statement claims that something is true for every object in a stated set or domain. It often contains words such as all, every, always or for any.

The domain is the set of permitted values or objects. For example, a statement might concern all integers, all positive real numbers or all triangles. The domain is an essential part of the statement.

What is a counterexample?

A counterexample is a particular example for which a universal statement is false.

Definition

Disproof by counterexample

To disprove a universal conjecture by counterexample, you give one value or object that satisfies the conditions of the conjecture but does not satisfy its conclusion.

Suppose someone conjectures:

For every real number $x$ , $x2≥xx^2\geq x$ .

Choose $x=12x=\frac12$ . Then

x2=(12)2=14<12=x.x^2=\left(\frac12\right)^2=\frac14<\frac12=x.

This one value disproves the conjecture.

Example

Disproving an inequality

Disprove the conjecture: “For every positive real number $x$ , $x+1x>2x+\frac1x>2$ .”

The conjecture concerns every positive real number, so a counterexample must satisfy $x > 0$ .
Equality is a natural boundary to investigate. Taking $x = 1$ gives
$x+1x=1+11=2.x+\frac1x=1+\frac11=2.$
The conjecture requires the expression to be strictly greater than 2, but for $x = 1$ it equals 2. Therefore $x = 1$ is a counterexample, so the conjecture is false.

Common Mistake

Ignoring strict inequalities

The symbols $>$ and $≥\geq$ have different meanings. If a conjecture claims an expression is greater than 2, a case where it equals 2 is enough to disprove it.

Why one example is enough

A universal statement promises that there are no exceptions. Therefore, finding just one genuine exception shows that the promise is false.

Think of the logical structure:

Conjecture: the property holds for every permitted case.
Counterexample: here is a permitted case where the property fails.
Conclusion: the conjecture is false.

However, checking examples for which the conjecture works does not prove it. Even a very large number of successful tests leaves open the possibility that a later case will fail.

Example

Testing a prime-number conjecture

Consider the conjecture: “For every non-negative integer $n$ , the number $n^2+n+41$ is prime.”

Small values produce primes, but this does not prove the conjecture. To challenge the expression, choose $n = 41$ , because this may introduce a factor of 41.
Substitute $n = 41$ :
$n2+n+41=412+41+41=412+2⋅41=41(41+2)=41⋅43.\begin{aligned} n^2+n+41 &=41^2+41+41\\ &=41^2+2\cdot41\\ &=41(41+2)\\ &=41\cdot43. \end{aligned}$
Since $41⋅4341\cdot43$ has factors other than 1 and itself, it is composite rather than prime. Therefore $n = 41$ is a counterexample, and the conjecture is false.

Key Idea

Evidence is not proof

A thousand examples that work do not prove a universal conjecture, but one valid example that fails disproves it.

The counterexample must fit the conditions

A proposed counterexample is only valid if it belongs to the conjecture’s stated domain and satisfies all its assumptions.

Consider the conjecture:

For every positive integer $n$ , the number $n^2+n$ is positive.

Using $n = - 1$ does not disprove this claim. Although

1)^2+(-1)=0,

the value $- 1$ is not a positive integer, so it is outside the domain.

Example

Checking the domain of a counterexample

Disprove the conjecture: “If $n$ is an integer, then $n2\frac{n}{2}$ is not an integer.”

The condition requires $n$ to be an integer, so choose an even integer such as $n = 6$ .
Substitute this value:
$n2=62=3.\frac{n}{2}=\frac62=3.$
The result 3 is an integer, contradicting the conclusion that $n2\frac n2$ is not an integer. Since $n = 6$ satisfies the original condition, it is a valid counterexample.

Common Mistake

Using an inadmissible value

Always check the domain before claiming a counterexample. For instance, a negative number cannot disprove a statement made only about positive numbers.

How to find a counterexample

Random testing can work, but a more systematic search is usually quicker.

Test boundary and special cases

Useful values often include:

zero and one;
negative values;
fractions between zero and one;
values where equality occurs;
the smallest permitted integer;
even and odd integers;
shapes with extra symmetry, such as squares or equilateral triangles.

These cases often expose an assumption that was hidden in the conjecture.

Example

Using a fraction to challenge a power claim

Disprove the conjecture: “Squaring a positive number always makes it larger.”

For values greater than 1, squaring does make the number larger, so investigate the different region $0 < x < 1$ .
Choose $x=12x=\frac12$ , which is positive, and calculate
$x2=(12)2=14.x^2=\left(\frac12\right)^2=\frac14.$
Since $14<12\frac14<\frac12$ , squaring has made this positive number smaller. Therefore $x=12x=\frac12$ is a counterexample and the conjecture is false.

Tip

Break the hidden assumption

Ask where the apparent pattern might change. In algebra, check zero, negative values and fractions; in number problems, check parity, factors and the ends of the stated range.

Negate the conclusion

To search effectively, identify exactly what would need to happen for the conclusion to fail.

For example, to disprove “the sum of any two prime numbers is even”, you need two prime numbers whose sum is not even, meaning their sum must be odd. Since 2 is the only even prime, it is a promising special case.

Example

Finding an odd sum of primes

Disprove the conjecture: “The sum of any two prime numbers is even.”

For a sum to be odd, one addend must be even and the other odd. The only even prime is 2, so pair it with an odd prime such as 3.
Both 2 and 3 satisfy the condition that they are prime, and their sum is
$2 + 3 = 5.$
The result 5 is odd, not even. Therefore the primes 2 and 3 form a counterexample, so the conjecture is false.

Writing a complete disproof

A counterexample should be accompanied by enough working to show why it contradicts the conjecture. A strong response has three parts:

State a permitted value or object.
Show that the conjecture’s conclusion fails for it.
Conclude explicitly that the conjecture is false.

Avoid writing only “false” or listing a number without explanation. The examiner needs to see that your example meets the conditions and contradicts the stated conclusion.

Common Mistake

Not every claim is universal

A single counterexample disproves a claim of the form “every object has this property”. It does not disprove an existence claim such as “there is an integer with this property”; one successful example would actually prove that existence claim.

Exam technique

In the exam

Identify the domain and every condition that your counterexample must satisfy.
Target the conclusion: decide exactly what result would make it false.
Try special cases such as zero, one, negative values, fractions, boundary values, even integers or odd integers.
Substitute or calculate clearly, then state: “Therefore this is a counterexample, so the conjecture is false.”
Do not waste time trying to prove a universal conjecture once you have found one valid counterexample.

Self review

Check yourself

Why can one counterexample disprove a universal statement, while many successful examples cannot prove it?
How would you disprove the conjecture that $x^3>x$ for every positive real number $x$ ?
What must you check before using a proposed value as a counterexample?

What you'll learn

What conjectures and counterexamples are.
Why one counterexample can disprove a universal claim.
How to search efficiently for a useful counterexample.
How to write a complete, logically sound disproof.

Conjectures and proof

A conjecture is a mathematical statement that is believed to be true but has not yet been proved. It may arise from noticing a pattern, testing examples or considering a mathematical model.

For example, after calculating

12+1+41=43,22+2+41=47,32+3+41=53,1^2+1+41=43,\qquad 2^2+2+41=47,\qquad 3^2+3+41=53,

you might conjecture that $n^2+n+41$ is prime for every positive integer $n$ .

Checking several examples can provide evidence for a conjecture, but evidence is not the same as proof. A proof must show that the statement is true in every case covered by the conjecture.

Definition

Universal statement

A universal statement claims that something is true for every object in a stated set or domain. It often contains words such as all, every, always or for any.

What is a counterexample?

A counterexample is a particular example for which a universal statement is false.

Definition

Disproof by counterexample

To disprove a universal conjecture by counterexample, you give one value or object that satisfies the conditions of the conjecture but does not satisfy its conclusion.

Suppose someone conjectures:

For every real number $x$ , $x2≥xx^2\geq x$ .

Choose $x=12x=\frac12$ . Then

x2=(12)2=14<12=x.x^2=\left(\frac12\right)^2=\frac14<\frac12=x.

This one value disproves the conjecture.

Example

Disproving an inequality

Disprove the conjecture: “For every positive real number $x$ , $x+1x>2x+\frac1x>2$ .”

The conjecture concerns every positive real number, so a counterexample must satisfy $x > 0$ .
Equality is a natural boundary to investigate. Taking $x = 1$ gives
$x+1x=1+11=2.x+\frac1x=1+\frac11=2.$
The conjecture requires the expression to be strictly greater than 2, but for $x = 1$ it equals 2. Therefore $x = 1$ is a counterexample, so the conjecture is false.

Common Mistake

Ignoring strict inequalities

The symbols $>$ and $≥\geq$ have different meanings. If a conjecture claims an expression is greater than 2, a case where it equals 2 is enough to disprove it.

Why one example is enough

A universal statement promises that there are no exceptions. Therefore, finding just one genuine exception shows that the promise is false.

Think of the logical structure:

Conjecture: the property holds for every permitted case.
Counterexample: here is a permitted case where the property fails.
Conclusion: the conjecture is false.

However, checking examples for which the conjecture works does not prove it. Even a very large number of successful tests leaves open the possibility that a later case will fail.

Example

Testing a prime-number conjecture

Consider the conjecture: “For every non-negative integer $n$ , the number $n^2+n+41$ is prime.”

Small values produce primes, but this does not prove the conjecture. To challenge the expression, choose $n = 41$ , because this may introduce a factor of 41.
Substitute $n = 41$ :
$n2+n+41=412+41+41=412+2⋅41=41(41+2)=41⋅43.\begin{aligned} n^2+n+41 &=41^2+41+41\\ &=41^2+2\cdot41\\ &=41(41+2)\\ &=41\cdot43. \end{aligned}$
Since $41⋅4341\cdot43$ has factors other than 1 and itself, it is composite rather than prime. Therefore $n = 41$ is a counterexample, and the conjecture is false.

Key Idea

Evidence is not proof

A thousand examples that work do not prove a universal conjecture, but one valid example that fails disproves it.

The counterexample must fit the conditions

A proposed counterexample is only valid if it belongs to the conjecture’s stated domain and satisfies all its assumptions.

Consider the conjecture:

For every positive integer $n$ , the number $n^2+n$ is positive.

Using $n = - 1$ does not disprove this claim. Although

1)^2+(-1)=0,

the value $- 1$ is not a positive integer, so it is outside the domain.

Example

Checking the domain of a counterexample

Disprove the conjecture: “If $n$ is an integer, then $n2\frac{n}{2}$ is not an integer.”

The condition requires $n$ to be an integer, so choose an even integer such as $n = 6$ .
Substitute this value:
$n2=62=3.\frac{n}{2}=\frac62=3.$
The result 3 is an integer, contradicting the conclusion that $n2\frac n2$ is not an integer. Since $n = 6$ satisfies the original condition, it is a valid counterexample.

Common Mistake

Using an inadmissible value

Always check the domain before claiming a counterexample. For instance, a negative number cannot disprove a statement made only about positive numbers.

How to find a counterexample

Random testing can work, but a more systematic search is usually quicker.

Test boundary and special cases

Useful values often include:

zero and one;
negative values;
fractions between zero and one;
values where equality occurs;
the smallest permitted integer;
even and odd integers;
shapes with extra symmetry, such as squares or equilateral triangles.

These cases often expose an assumption that was hidden in the conjecture.

Example

Using a fraction to challenge a power claim

Disprove the conjecture: “Squaring a positive number always makes it larger.”

For values greater than 1, squaring does make the number larger, so investigate the different region $0 < x < 1$ .
Choose $x=12x=\frac12$ , which is positive, and calculate
$x2=(12)2=14.x^2=\left(\frac12\right)^2=\frac14.$
Since $14<12\frac14<\frac12$ , squaring has made this positive number smaller. Therefore $x=12x=\frac12$ is a counterexample and the conjecture is false.

Tip

Break the hidden assumption

Ask where the apparent pattern might change. In algebra, check zero, negative values and fractions; in number problems, check parity, factors and the ends of the stated range.

Negate the conclusion

To search effectively, identify exactly what would need to happen for the conclusion to fail.

Example

Finding an odd sum of primes

Disprove the conjecture: “The sum of any two prime numbers is even.”

For a sum to be odd, one addend must be even and the other odd. The only even prime is 2, so pair it with an odd prime such as 3.
Both 2 and 3 satisfy the condition that they are prime, and their sum is
$2 + 3 = 5.$
The result 5 is odd, not even. Therefore the primes 2 and 3 form a counterexample, so the conjecture is false.

Writing a complete disproof

A counterexample should be accompanied by enough working to show why it contradicts the conjecture. A strong response has three parts:

State a permitted value or object.
Show that the conjecture’s conclusion fails for it.
Conclude explicitly that the conjecture is false.

Avoid writing only “false” or listing a number without explanation. The examiner needs to see that your example meets the conditions and contradicts the stated conclusion.

Common Mistake

Not every claim is universal

Exam technique

In the exam

Identify the domain and every condition that your counterexample must satisfy.
Target the conclusion: decide exactly what result would make it false.
Try special cases such as zero, one, negative values, fractions, boundary values, even integers or odd integers.
Substitute or calculate clearly, then state: “Therefore this is a counterexample, so the conjecture is false.”
Do not waste time trying to prove a universal conjecture once you have found one valid counterexample.

Self review

Check yourself

Why can one counterexample disprove a universal statement, while many successful examples cannot prove it?
How would you disprove the conjecture that $x^3>x$ for every positive real number $x$ ?
What must you check before using a proposed value as a counterexample?

What you'll learn

Conjectures and proof

What is a counterexample?

Why one example is enough

The counterexample must fit the conditions

How to find a counterexample

Test boundary and special cases

Negate the conclusion

Writing a complete disproof

Review 1.1.2 Disproof by counter example by teaching Genie

1.1.2 Disproof by counter example

What you'll learn

Conjectures and proof

What is a counterexample?

Why one example is enough

The counterexample must fit the conditions

How to find a counterexample

Test boundary and special cases

Negate the conclusion

Writing a complete disproof

Review 1.1.2 Disproof by counter example by teaching Genie

1.1.2 Disproof by counter example

What you'll learn

Conjectures and proof

What is a counterexample?

Why one example is enough

The counterexample must fit the conditions

How to find a counterexample

Test boundary and special cases

Negate the conclusion

Writing a complete disproof

Review 1.1.2 Disproof by counter example by teaching Genie

1.1.2 Disproof by counter example Revision Guide

1.1.2 Disproof by counter example

What you'll learn

Conjectures and proof

What is a counterexample?

Why one example is enough

The counterexample must fit the conditions

How to find a counterexample

Test boundary and special cases

Negate the conclusion

Writing a complete disproof

Review 1.1.2 Disproof by counter example by teaching Genie

1.1.2 Disproof by counter example Revision Guide