1.4 Disproof by counter example — Learn

Prerequisites: substituting a value into an expression; factors and prime numbers; the meaning of "for all" from 1.1; the limit of a finite check from 1.3.

By the end of this section you can: disprove a "for all" statement with one counter example; choose values to test with a reason; write a disproof in clear sentences; and explain why a counter example settles a "for all" claim completely.

1. One value is enough

A "for all" statement claims that every case works. To disprove it, you need one case that fails. That case is a counter example.

A disproof has three parts:

1. name a value that satisfies the assumption;
2. show that the conclusion fails for that value;
3. state that the statement is false.

Notice what a counter example does not do. It does not show that the opposite statement is true. "Every prime is odd" is false, because $2$ is an even prime, but that does not make "every prime is even" true.

2. A first counter example

Example 1. Disprove the claim "For every whole number $n$, the value of $n^2 + n + 41$ is prime".

Look for a value of $n$ that makes the expression factorise. The constant is $41$, so try $n = 40$. Then

Notice that $40 + 41 = 81 = 2 \times 40 + 1$, so the last two terms can be written as $2n + 1$ with $n = 40$:

Therefore

The value at $n = 40$ is $1681$, which has the factor $41$ and is not prime. The claim is false.

3. Where counter examples hide

Example 2. Disprove the claim "If $n$ is prime, then $2^n - 1$ is prime".

The claim fails at $n = 11$. First,

Now factorise $2047$:

Since $2047$ is a product of two numbers greater than $1$, it is not prime. The value $n = 11$ is prime, so the assumption holds and the conclusion fails. The claim is false.

Example 3. Disprove the claim "For every real number $x$, $x^2 > x$".

Test a number between $0$ and $1$. Take $x = \frac{1}{2}$. Then

Now

so $x^2 > x$ is false. The value $x = \frac{1}{2}$ satisfies the assumption (it is a real number), and the conclusion fails. The claim is false.

Example 4. Disprove the converse of a true statement.

The statement "If $n$ is a multiple of $6$, then $n$ is a multiple of $12$" is false. Take $n = 6$:

The number $6$ is a multiple of $6$, so the assumption holds, but it leaves remainder $6$ on division by $12$, so the conclusion fails. The claim is false.

Watch out. A value is a counter example only if it satisfies the assumption.

Suppose you want to disprove "If $n$ is even, then $n^2$ is even". Testing $n = 3$ tells you nothing, because $3$ is not even. The assumption is part of the claim, and a counter example must fit inside it.

Re-check: for each candidate value, write down two lines — "the assumption holds, because …" and "the conclusion fails, because …". If either line is missing, the value is not yet a counter example.

Progress check

Return to the chapter problem. Claim B says that $n^2 + n + 41$ is prime for every whole number $n$.

1. State the counter example in the form "the assumption holds, but the conclusion fails".
2. Why does a single value settle Claim B for every whole number?
3. Claim A says that $n^2 + n$ is even for every whole number $n$. Can a counter example disprove Claim A? What would you need to prove it instead?

You should now be able to

• disprove a "for all" statement with one counter example;
• check that a counter example satisfies the assumption;
• look for counter examples at boundaries, in fractions, and at the value built into a formula;
• explain why a counter example disproves a claim but does not prove its opposite.