What are the main parts of the structure of a proof?

State what is to be proved, state assumptions, give logical steps, and write a clear conclusion.

What is proof by exhaustion?

Checking every possible case and showing that the statement is true in each one.

When can proof by exhaustion be used?

Only when there is a finite number of cases that can all be listed and checked.

What do you write at the end of a proof by exhaustion?

A conclusion saying the statement is true because it holds in every case.

Why does checking n=1 to 5 not prove a statement for all positive integers?

There are infinitely many other cases that have not been checked.

Why must every case be shown, not just the answers?

So that the proof is complete and each step can be seen to be justified.

Structure of proof and proof by exhaustionEdexcel International A Level Maths: Flashcards

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What is a mathematical proof?

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What is a mathematical proof?: A logical argument from given assumptions, through justified steps, to a conclusion.
What are the main parts of the structure of a proof?: State what is to be proved, state assumptions, give logical steps, and write a clear conclusion.
What is proof by exhaustion?: Checking every possible case and showing that the statement is true in each one.
When can proof by exhaustion be used?: Only when there is a finite number of cases that can all be listed and checked.
What do you write at the end of a proof by exhaustion?: A conclusion saying the statement is true because it holds in every case.
If one case fails in an exhaustive check, what does that show?: The statement is false.
Why does checking $n=1$ to $5$ not prove a statement for all positive integers?: There are infinitely many other cases that have not been checked.
$x,y$ are odd positive integers less than $7$ . What values can each take?: $1$ , $3$ or $5$
List the possible values of $x+y$ for odd positive integers less than $7$ .: $2,4,6,8,10$ , all divisible by $2$
Evaluate $n^2+n$ for $n=1,2,3,4,5$ .: $2,6,12,20,30$
Why must every case be shown, not just the answers?: So that the proof is complete and each step can be seen to be justified.
$n^2+n+11$ for $n=10$ : is it prime?: No. It is $121=11^2$ .

Exam questions on Structure of proof and proof by exhaustion

A statement S is made: for every integer $n$ with $1\le n\le5$ , the value of $n^2+n$ is even.
Prove that S is true by exhaustion.2 marks
$x$ and $y$ are odd positive integers, each less than 7.
Prove that $x+y$ is divisible by $2$ for every possible pair.2 marks
Let $f(n)=n^3-n$ , where $n$ is an integer.
Use proof by exhaustion to show that $f(n)$ is divisible by $6$ for every integer $n$ with $2\le n\le6$ .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).