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Proof I

Proof by exhaustion

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Further kinematics

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Functions of time: Displacement, velocity, acceleration

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Constant acceleration

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Constant acceleration: Deriving formulae

Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

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Quantities and units in mechanics

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Normal distribution

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The normal distribution: Finding probabilities

The inverse normal distribution function

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Finding the mean and standard deviation

Normal approximation to the binomial distribution

Hypothesis testing with the normal distribution

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Hypothesis testing I

Hypothesis testing

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Binomial distribution

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Venn diagrams

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Vectors II

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Applications to mechanics: 3D vectors

Numerical methods

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Numerical methods: Applications to modelling

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Integrating standard functions

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Differentiation II

Differentiating sin x and cos x

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The chain rule

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Parametric differentiation

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Second derivatives: Concave and convex functions

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Sequences revision

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Binomial expansion II

Binomial expansion: (1+x)^n

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Parametric equations

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The modulus function

y = |f(x)| and y = f(|x|)

Combining transformations

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Proof II

Proof by contradiction

Criticising proofs

Vectors I

Vectors

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Magnitude and direction of a vector

Position vectors and displacement vectors

Solving geometric problems

Modelling with vectors

Integration I

Integrating x^n

Indefinite integrals

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Differentiation I

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Differentiation from first principles

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Gradients, tangents and normals

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Binomial expansion I

Pascal's triangle

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Binomial expansion

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Midpoints and perpendicular bisectors

Equation of a circle

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Direct and inverse proportion

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Equations and inequalities

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Set notation and interval notation

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Indices and surds

Laws of indices

Negative and fractional indices

Surds

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Proof I

Mathematical structure and arguments

Disproof by counterexample

Proof by deduction

Proof by exhaustion

Explainer Video

Tutor: Bilal

Summary

Proof by exhaustion

In a nutshell

Proof by exhaustion is a way of proving a statement that involves either testing every possible option or testing individual cases.

Testing every possible option

The most basic form of proof by exhaustion is to go through every single number and prove that all of them fit the claim.

Example 1

Prove that the square of every prime number between $10$ and $20$ ends in either a $1$ or a $9$ .

Proof by exhaustion is reasonable to use here because there are not many numbers to check.

List all the primes between $10$ and $20$ :

$11,13,17,19$ 

Square all of these primes:

$11^2=121\\13^2=169\\17^2=289\\19^2=361$

Hence, the square of all primes between these two numbers ends in a one or nine - the statement is correct.

Splitting up into cases

Another way to use proof by exhaustion is to split up the statement into two or more cases and address each case individually. This is helpful when proving more general cases.

Example 2

Prove that the square of any number is always a multiple of $3$ or $1$ more than a multiple of $3$ .

Because the claim mentions multiples of $3$ , split the cases into whether or not the number is a multiple of $3$ .

A number can either be a multiple of $3$ , $1$ more than a multiple of $3$ or $2$ more than a multiple of $3$ .

Write these algebraically:

$3n,3n+1,3n+2$

Work with each case separately.

Case 1: $3n$

$(3n)^2=9n^2=3(3n^2)$

So, this is a multiple of $3$ .

Case 2: $3n+1$

$(3n+1)^2=9n^2+6n+1=3(3n^2+2n)+1$ 

So, this is $1$ more than a multiple of $3$ .

Case 3: $3n+2$ 

$(3n+2)^2=9n^2+12n+4=3(3n^2+4n+1)+1$ 

So, this is also $1$ more than a multiple of $3$ .

Therefore, in every case, the square of a number is either a multiple of, or one more than a multiple of $3$ . - the statement is correct.

Learn with Basics

Length:

Unit 1

Algebraic terms

Unit 2

Algebraic proof - Higher

Jump Ahead

Optional

Unit 3

Proof by exhaustion

Final Test

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FAQs - Frequently Asked Questions

What are the two types of proof by exhaustion?

The two types of proof by exhaustion are: - Testing every single number. - Testing each individual case.

What is proof by exhaustion?

Proof by exhaustion is a way of proving a statement that involves either testing every possible option or testing individual cases.