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Proof II

Criticising proofs

Criticising proofs

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Further kinematics

Vector methods with projectiles

Variable acceleration in one dimension

Differentiating vectors

Integrating vectors

Application of forces

Static particles

Modelling with statics

Friction with static particles

Statics of rigid bodies

Dynamics and inclined planes

Connected particles

Projectiles

Horizontal projection

Projectiles: Horizontal and vertical components

Projection at any angle

Equation of the trajectory of a projectile

Forces and friction

Resolving forces

Inclined planes

The coefficient of friction

Moments

Moments

Resultant moments

Equilibrium

Centre of mass

Tilting

Laminas

Forces and motion

Force diagrams

Forces as vectors

Forces and acceleration

Motion in two dimensions

Connected particles

Pulleys

Variable acceleration

Functions of time: Displacement, velocity, acceleration

Using differentiation to find velocity and acceleration

Maxima and minima problems

Using integration to find velocity and displacement

Constant acceleration formulae

Constant acceleration

Displacement-time graphs

Velocity-time graphs

Constant acceleration: Deriving formulae

Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

Modelling assumptions

Quantities and units in mechanics

Working with vectors

Normal distribution

The normal distribution

The normal distribution: Finding probabilities

The inverse normal distribution function

The standard normal distribution

Finding the mean and standard deviation

Normal approximation to the binomial distribution

Hypothesis testing with the normal distribution

Conditional probability

Set notation

Conditional probability

Conditional probability in Venn diagrams

Probability formulae

Conditional probability: Tree diagrams

Correlation and hypothesis testing

Exponential models

Measuring correlation

Hypothesis testing for zero correlation

Hypothesis testing I

Hypothesis testing

Hypothesis testing: Finding critical values

One-tailed tests

Two-tailed tests

Binomial distribution

Probability distributions

The binomial distribution

Cumulative probabilities

Probability

Calculating probabilities

Venn diagrams

Probability: Mutually exclusive and independent events

Probability: Tree diagrams

Representation of data

Outliers

Box plots

Cumulative frequency

Histograms

Comparing data sets

Correlation

Linear regression

Measures of location and spread

Measures of central tendency: Mean, median, mode

Measures of spread: Range

Variance and standard deviation

Data collection

Population and samples

Sampling techniques

Types of data

Vectors II

3D coordinates

Vectors in 3D

Solving geometric problems

Applications to mechanics: 3D vectors

Numerical methods

Locating roots

Iteration

Staircase and cobweb diagrams

Newton-Raphson method

Numerical methods: Applications to modelling

Integration II

Integrating standard functions

Integrating f(ax + b)

Integration with trigonometric identities

Reverse chain rule

Integration by substitution

Integration by parts

Integration using partial fractions

Finding areas using integration

The trapezium rule

Solving differential equations

Modelling with differential equations

Differentiation II

Differentiating sin x and cos x

Differentiating exponentials and logarithmic functions

The chain rule

The product rule

The quotient rule

Differentiating inverse functions

Differentiating trigonometric functions

Parametric differentiation

Implicit differentiation

Second derivatives: Concave and convex functions

Connected rates of change

Trigonometry and modelling

Addition formulae

Using the angle addition formulae

Double angle formulae

Solving trigonometric equations: Double angle formula

Simplifying a cos x + b sin x

Proving trigonometric identities

Modelling with trigonometric functions

Trigonometric functions

Secant, cosecant and cotangent

Graphs of sec x, cosec x and cot x

Solving equations with sec x, cosec x and cot x

Further trigonometric identities

Inverse trigonometric functions

Radians

Radian measure

Arc length

Areas of sectors and segments

Solving trigonometric equations

Small angle approximations

Sequences and series

Sequences revision

Arithmetic sequences

Arithmetic series

Geometric sequences

Geometric series

Sum to infinity

Sigma notation

Modelling with sequences and series

Binomial expansion II

Binomial expansion: (1+x)^n

Binomial expansion: (a+bx)^n

Binomial expansion with partial fractions

Binomial expansion of compound expressions

Parametric equations

Parametric equations

Using trigonometric identities

Sketching parametric curves

Points of intersection of parametric curves

Modelling with parametric equations

Functions

Functions and mappings

Composite functions

Inverse functions

The modulus function

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

Combining transformations

Solving modulus equations

Modulus inequalities

Algebraic fractions

Operations with algebraic fractions

Partial fractions

Repeated factors

Algebraic division

Proof II

Proof by contradiction

Criticising proofs

Vectors I

Vectors

Representing vectors

Magnitude and direction of a vector

Position vectors and displacement vectors

Solving geometric problems

Modelling with vectors

Integration I

Integrating x^n

Indefinite integrals

Finding functions by integrating

Definite integrals

Area under a curve

Area under the x-axis

Area between a curve and a line

Differentiation I

Finding the gradient of a curve

Differentiation from first principles

Differentiating x^n

Differentiating functions with multiple terms

Gradients, tangents and normals

Increasing and decreasing functions

Second order derivatives

Stationary points

Sketching gradient functions

Modelling with differentiation

Exponentials and logarithms

Exponential functions

y = e^x

Exponential modelling

Logarithms

Laws of logarithms

Solving equations using logarithms

Natural logarithms

Logarithms and non-linear data

Harder exponential modelling

Trigonometric equations

Angles in all four quadrants

Exact trigonometric values

Trigonometric identities

Simple trigonometric equations

Harder trigonometric equations

Equations and identities

Trigonometry

The cosine rule

The sine rule

Area of a triangle

Solving triangle problems

Trig graphs: sin x, cos x, tan x

Transforming trigonometric graphs

Binomial expansion I

Pascal's triangle

Factorial notation

Binomial expansion

Solving binomial problems

Binomial estimation

Circles

Midpoints and perpendicular bisectors

Equation of a circle

Intersection of straight lines and circles

Using tangent and chord properties

Circles and triangles

Intersection of two circles

Straight line graphs

y = mx + c

Equations of straight lines

Parallel and perpendicular lines

Length and area

Modelling with straight lines

Transformations

Translating graphs

Stretching and reflecting graphs

Transforming graphs

Sketching graphs

Cubic graphs

Quartic graphs

Reciprocal graphs

Points of intersection

Using the discriminant

Direct and inverse proportion

Polynomials

Functions

Expanding brackets

Factorising

Simplifying algebraic fractions

Polynomial division

The factor theorem

Equations and inequalities

Linear simultaneous equations

Quadratic simultaneous equations

Set notation and interval notation

Linear inequalities

Quadratic inequalities

Inequalities on graphs

Inequality regions

Quadratics

Solving quadratic equations

Completing the square

Quadratic graphs

Identifying the discriminant

Solving disguised quadratics

Indices and surds

Laws of indices

Negative and fractional indices

Surds

Rationalising the denominator

Proof I

Mathematical structure and arguments

Disproof by counterexample

Proof by deduction

Proof by exhaustion

Explainer Video

Tutor: Dylan

Summary

Criticising proofs

​​In a nutshell

You need to be able to follow the logical steps in a proof, and identify where mistakes have been made in the proofs of others.


​

Implication notation

Given statements AAA​ and BBB​, the following arrow symbols can be used to indicate ways in which they relate to each other:


​A ⟹ BA \implies BA⟹B​​

"AAA​ implies BBB​" - this means that if AAA​ is true, then BBB​ is also true.

A ⟸ BA \impliedby BA⟸B​​

"BBB implies AAA​" - this means that if BBB​ is true, then AAA​ is also true.

​A ⟺ BA \iff BA⟺B​​

"AAA​ if and only if BBB​" - AAA​ and BBB​ are logically equivalent, i.e. one is true if and only if the other is true.



Negating statements

Suppose you have a statement of the form: "for all xxx, P(x)P(x)P(x)" where P(x)P(x)P(x)​ is some statement dependent on xxx​. The negation of this statement would not be of the form "for all xxx​, ¬P(x)¬P(x)¬P(x)​" (where ¬P(x)¬P(x)¬P(x)​ is the negation of P(x)P(x)P(x)​). This is far stronger than is necessary to negate this; in fact, the negation would be "there exists an xxx such that ¬P(x)¬P(x)¬P(x)​", as either P(x)P(x)P(x)​ is true for any xxx​, or there exists an xxx​ such that ¬P(x)¬P(x)¬P(x)​.


​​Example 1

Identify the error in the following attempt to prove that the sum of all the proper divisors of any positive integer is not equal to the integer itself.

(A proper divisor of a positive integer is another integer which divides it that is not equal to the integer itself.)


The proof proceeds by contradiction; assume the negation of the statement, i.e. that the sum of the proper divisors of any proper integer is equal to the integer itself. Consider any prime number ppp​. Its only proper divisor is 111​, but p≠1p \ne 1p=1, so this is a contradiction, as you have assumed that the proper divisors of any number add to the number itself.


Therefore, the sum of the proper divisors of any integer is not equal to the integer itself.

​

The flaw in the proof comes at the point the negation is assumed: the negation of "the sum of all the proper divisors of any positive integer does not equal the integer itself" is "there exists an integer such that the sum of all its proper divisors does equal the integer". This statement cannot be used to derive a contradiction, as it is in fact true, e.g. 666, 282828, 496496496, and many more.


Therefore, the error in the attempted proof is incorrectly negating the statement while trying to derive a contradiction.


Example 2

Identify the error in the following proof that 1=01 = 01=0​:


Let x=1x = 1x=1​.


1=x ⟺ 1−x2=x−x2 ⟺ (1−x)(1+x)=x(1−x) ⟺ 1+x=x ⟺ 1=0\begin{aligned}1 &= x\\\iff1 - x^2 &= x - x^2\\\iff (1-x)(1+x) &= x(1-x)\\\iff1 + x &= x\\\iff 1 &= 0\end{aligned}1⟺1−x2⟺(1−x)(1+x)⟺1+x⟺1​=x=x−x2=x(1−x)=x=0​​


The error comes in dividing by 1−x1-x1−x - this is equal to 000, so you cannot divide by it.


Therefore, the error comes in dividing by zero.


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Expressions, equations, formulae, functions and identities

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Expressions, equations, formulae, functions and identities

Mathematical structure and arguments

Unit 2

Mathematical structure and arguments

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Criticising proofs

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Criticising proofs

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FAQs - Frequently Asked Questions

Does taking the negation of something twice leave it unchanged?

Yes.

Why is the negation of a "for all" statement a "there exists" statement?

Either P(x) is true for any x, or there exists an x such that ¬P(x).

What is the negation of "for all x, P(x)"?

The negation would be "there exists an x such that ¬P(x)".

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