Disproof by counter-exampleAQA A-Level Maths: Revision notes

A counter-example is a single case that satisfies the conditions of a statement but makes its conclusion false. It disproves a statement of the form "for all $n$" or "if ... then ...".

• One counter-example is enough to disprove a statement.
• No number of confirming examples proves a general statement.
• A statement that has been proved true can never have a counter-example. Example: "for every positive integer $n$, $n^2+n+41$ is prime". Values $n=1$ to $39$ all give primes, but $n=40$ gives $1681=41^2$, which is not prime.

Test small and special cases first:

• $n=0$, $n=1$, $n=2$ (the only even prime), and negative numbers if allowed;
• values on a boundary of an inequality, such as $x=0$ or $x=1$ for $x^2>x$;
• surds that cancel, such as $\sqrt2$ and $-\sqrt2$, or equal values, such as $\sqrt2\times\sqrt2=2$;
• larger values when small ones fail to break the claim, such as $n=11$ for "if $n$ is prime then $2^n-1$ is prime". Algebra can help: $x^2\le x$ means $x(x-1)\le0$, so every $x$ with $0\le x\le1$ is a counter-example to $x^2>x$.

A complete disproof has three parts:

1. State a specific counter-example.
2. Show that it satisfies the conditions of the statement.
3. Show that the conclusion is false for it. Example: to disprove "the sum of two primes is even", choose $2$ and $3$. Both are prime, but $2+3=5$ is odd. Example: to disprove "if $x$ and $y$ are irrational then $xy$ is irrational", choose $x=y=\sqrt2$. Both are irrational, but $xy=2$ is rational. Always show the working, such as $2^{11}-1=2047=23\times89$, rather than just writing the number.

If you cannot find a counter-example, that does not show the statement is true; you need a proof. The statement "if $n$ is odd then $n^2-1$ is a multiple of 8" is true: with $n=2k+1$, $n^2-1=4k(k+1)$, and $k(k+1)$ is even, so this is a multiple of 8, and no counter-example exists. Quick guide:

• looks false: search for a counter-example;
• looks true: prove it by deduction or exhaustion;
• if testing many cases finds no counter-example, that is evidence, not proof. Strong claims such as "for all" are the easiest to disprove; a claim that something exists needs one example to prove it.