Disproof by counter-exampleAQA A-Level Maths: Flashcards

What these 12 flashcards ask

What is a counter-example?
How many counter-examples disprove a statement?
Does finding no counter-example prove a statement?
Counter-example to 'the sum of two primes is even'?
Counter-example to 'n^3n^2 for all n'?
Counter-example to 'if n is prime then 2^n-1 is prime'?
Counter-example to 'x^2x'?
Counter-example to 'irrational plus irrational is irrational'?
Counter-example to 'irrational times irrational is irrational'?
Counter-example to 'n^2+n+41 is always prime'?
What three things should a disproof show?
Which prime is the usual exception to 'primes are odd'?

Exam questions on Disproof by counter-example

A student claims: 'For every positive integer $n$ , the number $n^2+n+41$ is prime.'
The student first tested $n=1,2,3,4,5$ and found a prime each time. Explain why this did not prove the claim, and what is needed to disprove it.2 marks
Consider two statements. Statement P: if $x$ and $y$ are irrational numbers then $x+y$ is irrational. Statement Q: if $x$ and $y$ are irrational numbers then $xy$ is irrational.
Give a counter-example to statement P in which $y\ne-x$ , and show that it is a counter-example.2 marks
Consider the statement: for every real number $x$ , $x^2>x$ .
Show that the statement is false by finding a counter-example and explaining clearly why it works.3 marks

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).