Can examples that agree with a statement prove it for all values?

No. Only a proof covering all cases can do that.

Disprove: 'n^2-n+1 is prime for all integers n\ge2'.

n=5: 21=3\times7, which is not prime.

Disprove: 'x^2x for all real x'.

x=\frac12: \frac14 is not greater than \frac12. (x=1 and x=0 also work.)

How do you show that a value such as 21 is not prime?

Write it as a product of two smaller whole numbers: 21=3\times7.

What must a counterexample satisfy?

The conditions of the statement, while making the statement itself false.

Which values are often good to test?

0, 1, negatives, fractions between 0 and 1, and the prime 2.

What should the last line of a disproof say?

A conclusion such as 'so the statement is false'.

Disproof by counter exampleEdexcel International A Level Maths: Flashcards

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What is a counterexample?

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What is a counterexample?: A single value, allowed by the conditions of a statement, for which the statement is false.
How many counterexamples are needed to disprove a statement?: One.
Can examples that agree with a statement prove it for all values?: No. Only a proof covering all cases can do that.
Disprove: ' $n^2-n+1$ is prime for all integers $n\ge2$ '.: $n=5$ : $21=3\times7$ , which is not prime.
Disprove: 'if $p$ is prime then $p+2$ is prime'.: $p=7$ : $9=3\times3$ is not prime.
Disprove: 'if $a^2=b^2$ then $a=b$ '.: $a=3$ , $b=-3$ : $9=9$ but $3\ne-3$ .
Disprove: ' $x^2>x$ for all real $x$ '.: $x=\frac12$ : $\frac14$ is not greater than $\frac12$ . ( $x=1$ and $x=0$ also work.)
Disprove: ' $2^n+1$ is prime for all positive integers $n$ '.: $n=3$ : $9=3\times3$ .
How do you show that a value such as $21$ is not prime?: Write it as a product of two smaller whole numbers: $21=3\times7$ .
What must a counterexample satisfy?: The conditions of the statement, while making the statement itself false.
Which values are often good to test?: $0$ , $1$ , negatives, fractions between $0$ and $1$ , and the prime $2$ .
What should the last line of a disproof say?: A conclusion such as 'so the statement is false'.

Exam questions on Disproof by counter example

A statement T is made: for all real numbers $x$ , $x^2>x$ .
Find a value of $x$ with $0<x<1$ and use it to show that T is false.2 marks
A student states: 'If $p$ is a prime number, then $p+2$ is also a prime number.'
Find a counterexample to the statement with $p>10$ . Show your working.2 marks
Consider the statement U: $n^2-n+1$ is a prime number for all integers $n\ge2$ .
Disprove U.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).