A logical argument from assumptions, through valid steps, to a conclusion that holds for every case covered.

Why is checking examples not a proof?

It covers only some cases; a proof must cover all of them.

How do you write any even integer? Any odd integer?

Even: 2k. Odd: 2k+1, where k is an integer.

Write the sum of two consecutive squares in the form 2(\ldots)+1.

n^2+(n+1)^2=2(n^2+n)+1, which is odd.

Prove n^2-6n+100 in one step.

(n-3)^2+1\geqslant1, since a square is at least 0.

What is proof by exhaustion?

Splitting into a finite number of cases and checking every one.

Why must n(n+1) be even?

One of two consecutive integers is even.

How do you prove the sum formula for an arithmetic series?

Write the series forwards and backwards and add: 2S=n(a+l).

What should the last line of a proof say?

The conclusion in words, linking back to the original claim.

What does \Rightarrow mean?

'implies': the line before leads to the next.

Proof by deduction, exhaustion and counter-exampleEdexcel A-Level Maths: Flashcards

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What is a proof?

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What is a proof?: A logical argument from assumptions, through valid steps, to a conclusion that holds for every case covered.
Why is checking examples not a proof?: It covers only some cases; a proof must cover all of them.
How do you write any even integer? Any odd integer?: Even: $2k$ . Odd: $2k+1$ , where $k$ is an integer.
Write two consecutive integers.: $n$ and $n+1$ .
Write the sum of two consecutive squares in the form $2(\ldots)+1$ .: $n^2+(n+1)^2=2(n^2+n)+1$ , which is odd.
Prove $n^2-6n+10>0$ in one step.: $(n-3)^2+1\geqslant1$ , since a square is at least 0.
What is proof by exhaustion?: Splitting into a finite number of cases and checking every one.
List the primes between 3 and 25.: $5,7,11,13,17,19,23$
How many counter-examples disprove a statement?: One is enough.
Give a counter-example to ' $n^2+n+41$ is prime for all $n$ '.: $n=40$ : $1681=41^2$ .
Why must $n(n+1)$ be even?: One of two consecutive integers is even.
How do you prove the sum formula for an arithmetic series?: Write the series forwards and backwards and add: $2S=n(a+l)$ .
What should the last line of a proof say?: The conclusion in words, linking back to the original claim.
What does $\Rightarrow$ mean?: 'implies': the line before leads to the next.

Exam questions on Proof by deduction, exhaustion and counter-example

A student claims that $n^2+n+41$ is a prime number for every positive integer $n$ .
Show that the claim is false for $n=41$ , and state what type of proof this is.2 marks
The function $f(n)=n^2-6n+10$ is defined for all real $n$ .
Hence prove that $n^2-6n+10$ is positive for all real $n$ .2 marks
Two consecutive integers are $n$ and $n+1$ , where $n$ is an integer.
Prove that the sum of the squares of the two integers is always odd.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).