Search
StudyWell
  • Home
  • Maths
    • AS Maths
    • A2 Maths
    • Pure Maths
    • Statistics
    • Mechanics
  • Study Resources
    • Questions by Topic
    • Past & Practice Papers
    • AS Pure Maths Videos
  • Shop
  • My Account
  • Home
  • Maths
    • AS Maths
    • A2 Maths
    • Pure Maths
    • Statistics
    • Mechanics
  • Study Resources
    • Questions by Topic
    • Past & Practice Papers
    • AS Pure Maths Videos
  • Shop
  • My Account

How to do Proof by Exhaustion – Examples & Videos

StudyWell > Proof – mathematical logic and reasoning > How to do Proof by Exhaustion – Examples & Videos

Proof by Exhaustion Notes

Proof by Exhaustion is the proof that something is true by showing that it is true for each and every case that could possibly be considered. This is also known as Proof by Cases– see Example 1. This is different from Proof by Deduction where we use algebraic symbols and construct logical arguments from known facts to show that something is true for all numbers. For the case of Proof by Exhaustion, we show that a statement is true for each number in consideration (or subsets of numbers).
Proof by Exhaustion also includes proof where numbers are split into a set of exhaustive categories and the statement is shown to be true for each category. Proof by Deduction can then be used within the categories – see Example 2.

Examples of Proof by Exhaustion

Example 1

Prove that $(n+1)^3\geq 3^n$ for $n\in{\mathbb N}, n\leq 4$.

Show Solution

We must show that $(n+1)^3$ is greater than $3n$ for all $n$ that is a natural number less than or equal to 4. The natural numbers less than or equal to 4 include 1, 2, 3, and 4. Hence, proving the above by exhaustion will involve showing that it is true for $n$ set equal to 1, 2, 3 and 4:

  1. For $n=1$, $(n+1)^3=(1+1)^3=2^3=8$ and $3^n=3^1=3$. Since, $8\geq 3$, the above is true when $n=1$.
  2. For $n=2$, $(n+1)^3=(2+1)^3=3^3=27$ and $3^n=3^2=9$. Since, $27\geq 9$, the above is true when $n=2$.
  3. For $n=3$, $(n+1)^3=(3+1)^3=4^3=64$ and $3^n=3^3=27$. Since, $64\geq 27$, the above is true when $n=3$.
  4. For $n=4$, $(n+1)^3=(4+1)^3=5^3=125$ and $3^n=3^4=81$. Since, $125\geq 81$, the above is true when $n=4$ and thus concluding the proof by exhaustion.

Note that $n=5$ is the first instance when the statement breaks down.

Example 2

Prove that every perfect cube number is a multiple of 9, one less than a multiple of 9 or one more than a multiple of 9.

Show Solution

Trick: Every integer n is a multiple of 3 ($n=3m$), one less than a multiple of 3 ($n=3m-1$) or one more than a multiple of 3 ($n=3m+1$). Consider $n=3m$, then

$n^3=(3m)^3=27m^3=9\times 3m^3$

which is a multiple of 9. Now consider $n=3m-1$, then

$\begin{array}{l}n^3&=&(3m-1)^3\\&=&27m^3-27m^2+9m-1\\&=&9\left(3m^3-3m^2+m\right)-1\end{array}$

which is one less than a multiple of 9. Finally consider $n=3m+1$, then

$\begin{array}{l}n^3&=&(3m+1)^3\\&=& 27m^3+27m^2+9m+1\\&=&9\left(3m^3+3m^2+m\right)+1\end{array}$

which is one more than a multiple of 9 and concluding the proof by exhaustion.

Exam-Style Proof Questions

proof by exhaustion

Download 16 Exam-Style Proof Questions.

More on Proof

  • See PROOF BY DEDUCTION
  • DISPROOF BY COUNTER-EXAMPLE
  • Go back to PURE MATHS
  • See QUESTIONS BY TOPIC
  • Go to PAST and PRACTICE PAPERS

Where does the word proof come from?

Proof by Exhaustion Videos

See more videos.

More Proof by Exhaustion videos.

About

StudyWell is a website for students studying A-Level Maths (or equivalent. course). We have lots of resources including A-Level content delivered in manageable bite-size pieces, practice papers, past papers, questions by topic, worksheets, hints, tips, advice and much, much more.

Quick Links

  • CONTACT US
  • REGISTER
  • Edexcel Exam Timetable
  • Edexcel Formula Booklet
  • Edexcel Grade Boundaries
  • Edexcel Large Data Set
  • Edexcel Specification

Top Pages

  • A2 Maths (second year of A-Level Maths)
  • AS Maths (first year of A-Level Mathematics)
  • Blog
  • My account
  • Practice Papers
  • Questions by Topic
  • Shop
  • Membership Levels

Useful Websites

  • DESMOS
  • GeoGebra
  • Maths Challenges
  • STEP papers
  • UCAS
  • Wolfram Alpha
  • Friend of StudyWell: Elite Locksmiths
Footer logo
Copyright © 2022 StudyWell
MENU logo
  • Home
  • Maths
    • AS Maths
    • A2 Maths
    • Pure Maths
    • Statistics
    • Mechanics
  • Study Resources
    • Questions by Topic
    • Past & Practice Papers
    • AS Pure Maths Videos
  • Shop
  • My Account