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OCR A Level Maths: Pure

Revision Notes

Home / A Level / Maths: Pure / OCR / Revision Notes / 1. Proof / 1.1 Proof / 1.1.2 Proof by Deduction


1.1.2 Proof by Deduction


Proof by Deduction

What is proof by deduction?

  • Proof by deduction is when a mathematical and logical argument is used to shows whether or not a result is true

How to do proof by deduction

Proof by Deduction Notes Diagram 1, Edexcel A Level Maths: Pure revision notes

 You may also need to:

  • Write multiples of n in the form kn for some integer k
  • Use algebraic techniques, showing logical steps of simplifying
  • Use correct mathematical notation

Sets of numbers

  • ℕ - the set of natural numbers
  • ℤ - the set of integers
  • ℚ - the set of quotients/rational numbers
  • ℝ - the set of real numbers

 Proof by Deduction Notes Diagram 2, Edexcel A Level Maths: Pure revision notes

Exam Tip

  • Try the result you are proving with a few values– use a sequence of them (eg 1, 2, 3)
  • This may help you see a pattern and spot what is going on. 

Worked Example

Proof by Deduction Example, IGCSE & GCSE Maths revision notes



  • 1. Proof
    • 1.1 Proof
      • 1.1.1 Language of Proof
        • 1.1.2 Proof by Deduction
          • 1.1.3 Proof by Exhaustion
            • 1.1.4 Disproof by Counter Example
            • 1.2 Proof by Contradiction
              • 1.2.1 Proof by Contradiction
            • 2. Algebra & Functions
              • 2.1 Laws of Indices & Surds
                • 2.1.1 Laws of Indices
                  • 2.1.2 Manipulating Surds
                    • 2.1.3 Surds - Rationalising the Denominator
                    • 2.2 Quadratics
                      • 2.2.1 Quadratic Graphs
                        • 2.2.2 Discriminants
                          • 2.2.3 Completing the square
                            • 2.2.4 Solving Quadratic Equations
                              • 2.2.5 Further Solving Quadratic Equations (Hidden Quadratics)
                              • 2.3 Simultaneous Equations
                                • 2.3.1 Linear Simultaneous Equations - Elimination
                                  • 2.3.2 Linear Simultaneous Equations - Substitution
                                    • 2.3.3 Quadratic Simultaneous Equations
                                    • 2.4 Inequalities
                                      • 2.4.1 Linear Inequalities
                                        • 2.4.2 Quadratic Inequalities
                                          • 2.4.3 Inequalities on Graphs
                                          • 2.5 Polynomials
                                            • 2.5.1 Expanding Brackets
                                              • 2.5.2 Polynomial Division
                                                • 2.5.3 Factor Theorem
                                                  • 2.5.4 Factorisation
                                                  • 2.6 Rational Expressions
                                                    • 2.6.1 Rational Expressions
                                                      • 2.6.2 Improper Algebraic Fractions
                                                      • 2.7 Graphs of Functions
                                                        • 2.7.1 Sketching Polynomials
                                                          • 2.7.2 Reciprocal Graphs - Sketching
                                                            • 2.7.3 Solving Equations Graphically
                                                              • 2.7.4 Proportional Relationships
                                                              • 2.8 Functions
                                                                • 2.8.1 Language of Functions
                                                                  • 2.8.2 Composite Functions
                                                                    • 2.8.3 Inverse Functions
                                                                      • 2.8.4 Modulus Functions - Sketching Graphs
                                                                        • 2.8.5 Modulus Functions - Solving Equations
                                                                        • 2.9 Transformations of Functions
                                                                          • 2.9.1 Translations
                                                                            • 2.9.2 Stretches
                                                                              • 2.9.3 Reflections
                                                                              • 2.10 Combinations of Transformations
                                                                                • 2.10.1 Combinations of Transformations
                                                                                • 2.11 Partial Fractions
                                                                                  • 2.11.1 Linear Denominators
                                                                                    • 2.11.2 Squared Linear Denominators
                                                                                    • 2.12 Modelling with Functions
                                                                                      • 2.12.1 Modelling with Functions
                                                                                      • 2.13 Further Modelling with Functions
                                                                                        • 2.13.1 Further Modelling with Functions
                                                                                      • 3. Coordinate Geometry
                                                                                        • 3.1 Equation of a Straight Line
                                                                                          • 3.1.1 Basic Coordinate Geometry
                                                                                            • 3.1.2 Parallel & Perpendicular Gradients
                                                                                              • 3.1.3 Equation of a Straight Line
                                                                                                • 3.1.4 Modelling with Straight Lines
                                                                                                • 3.2 Circles
                                                                                                  • 3.2.1 Equation of a Circle
                                                                                                    • 3.2.2 Finding the Centre & Radius
                                                                                                      • 3.2.3 Bisection of Chords
                                                                                                        • 3.2.4 Angle in a Semicircle
                                                                                                          • 3.2.5 Radius & Tangent
                                                                                                        • 4. Sequences and Series
                                                                                                          • 4.1 Binomial Expansion
                                                                                                            • 4.1.1 Binomial Expansion
                                                                                                            • 4.2 General Binomial Expansion
                                                                                                              • 4.2.1 General Binomial Expansion
                                                                                                                • 4.2.2 General Binomial Expansion - Subtleties
                                                                                                                  • 4.2.3 General Binomial Expansion - Multiple
                                                                                                                    • 4.2.4 Approximating Values
                                                                                                                    • 4.3 Arithmetic Sequences & Series
                                                                                                                      • 4.3.1 Arithmetic Sequences
                                                                                                                        • 4.3.2 Arithmetic Series
                                                                                                                        • 4.4 Geometric Sequences & Series
                                                                                                                          • 4.4.1 Geometric Sequences
                                                                                                                            • 4.4.2 Geometric Series
                                                                                                                            • 4.5 Sequences & Series
                                                                                                                              • 4.5.1 Language of Sequences & Series
                                                                                                                                • 4.5.2 Sigma Notation
                                                                                                                                  • 4.5.3 Recurrence Relations
                                                                                                                                  • 4.6 Modelling with Sequences & Series
                                                                                                                                    • 4.6.1 Modelling with Sequences & Series
                                                                                                                                  • 5. Trigonometry
                                                                                                                                    • 5.1 Basic Trigonometry
                                                                                                                                      • 5.1.1 Trigonometry - Definitions
                                                                                                                                        • 5.1.2 Right-Angled Triangles
                                                                                                                                          • 5.1.3 Non-Right-Angled Triangles
                                                                                                                                          • 5.2 Trigonometric Functions
                                                                                                                                            • 5.2.1 Graphs of Trigonometric Functions
                                                                                                                                              • 5.2.2 Transformations of Trigonometric Functions
                                                                                                                                              • 5.3 Trigonometric Equations
                                                                                                                                                • 5.3.1 Trigonometry - Simple Identities
                                                                                                                                                  • 5.3.2 Linear Trigonometric Equations
                                                                                                                                                    • 5.3.3 Quadratic Trigonometric Equations
                                                                                                                                                      • 5.3.4 Strategy for Trigonometric Equations
                                                                                                                                                      • 5.4 Radian Measure
                                                                                                                                                        • 5.4.1 Radian Measure
                                                                                                                                                          • 5.4.2 Trigonometry Exact Values
                                                                                                                                                            • 5.4.3 Small Angle Approximations
                                                                                                                                                            • 5.5 Reciprocal & Inverse Trigonometric Functions
                                                                                                                                                              • 5.5.1 Reciprocal Trig Functions - Definitions
                                                                                                                                                                • 5.5.2 Reciprocal Trig Functions - Graphs
                                                                                                                                                                  • 5.5.3 Trigonometry - Further Identities
                                                                                                                                                                    • 5.5.4 Inverse Trig Functions
                                                                                                                                                                    • 5.6 Compound & Double Angle Formulae
                                                                                                                                                                      • 5.6.1 Compound Angle Formulae
                                                                                                                                                                        • 5.6.2 Double Angle Formulae
                                                                                                                                                                          • 5.6.3 R addition formulae Rcos Rsin etc
                                                                                                                                                                          • 5.7 Further Trigonometric Equations
                                                                                                                                                                            • 5.7.1 Strategy for Further Trigonometric Equations
                                                                                                                                                                            • 5.8 Trigonometric Proof
                                                                                                                                                                              • 5.8.1 Trigonometric Proof
                                                                                                                                                                              • 5.9 Modelling with Trigonometric Functions
                                                                                                                                                                                • 5.9.1 Modelling with Trigonometric Functions
                                                                                                                                                                              • 6. Exponentials & Logarithms
                                                                                                                                                                                • 6.1 Exponential & Logarithms
                                                                                                                                                                                  • 6.1.1 Exponential Functions
                                                                                                                                                                                    • 6.1.2 Logarithmic Functions
                                                                                                                                                                                      • 6.1.3 "e"
                                                                                                                                                                                        • 6.1.4 Derivatives of Exponential Functions
                                                                                                                                                                                        • 6.2 Laws of Logarithms
                                                                                                                                                                                          • 6.2.1 Laws of Logarithms
                                                                                                                                                                                            • 6.2.2 Exponential Equations
                                                                                                                                                                                            • 6.3 Modelling with Exponentials & Logarithms
                                                                                                                                                                                              • 6.3.1 Exponential Growth & Decay
                                                                                                                                                                                                • 6.3.2 Using Exps & Logs in Modelling
                                                                                                                                                                                                  • 6.3.3 Using Log Graphs in Modelling
                                                                                                                                                                                                • 7. Differentiation
                                                                                                                                                                                                  • 7.1 Differentiation
                                                                                                                                                                                                    • 7.1.1 Definition of Gradient
                                                                                                                                                                                                      • 7.1.2 First Principles Differentiation
                                                                                                                                                                                                        • 7.1.3 Differentiating Powers of x
                                                                                                                                                                                                        • 7.2 Applications of Differentiation
                                                                                                                                                                                                          • 7.2.1 Gradients, Tangents & Normals
                                                                                                                                                                                                            • 7.2.2 Increasing & Decreasing Functions
                                                                                                                                                                                                              • 7.2.3 Second Order Derivatives
                                                                                                                                                                                                                • 7.2.4 Stationary Points & Turning Points
                                                                                                                                                                                                                  • 7.2.5 Sketching Gradient Functions
                                                                                                                                                                                                                    • 7.2.6 Modelling with Differentiation inc. Optimisation
                                                                                                                                                                                                                    • 7.3 Further Differentiation
                                                                                                                                                                                                                      • 7.3.1 First Principles Differentiation - Trigonometry
                                                                                                                                                                                                                        • 7.3.2 Differentiating Other Functions (Trig, ln & e etc)
                                                                                                                                                                                                                          • 7.3.3 Chain Rule
                                                                                                                                                                                                                            • 7.3.4 Product Rule
                                                                                                                                                                                                                              • 7.3.5 Quotient Rule
                                                                                                                                                                                                                                • 7.3.6 Differentiating Reciprocal and Inverse Trig Functions
                                                                                                                                                                                                                                • 7.4 Further Applications of Differentiation
                                                                                                                                                                                                                                  • 7.4.1 Applications of the Second Derivative
                                                                                                                                                                                                                                    • 7.4.2 Points of Inflection
                                                                                                                                                                                                                                      • 7.4.3 Connected Rates of Change
                                                                                                                                                                                                                                      • 7.5 Implicit Differentiation
                                                                                                                                                                                                                                        • 7.5.1 Implicit Differentiation
                                                                                                                                                                                                                                      • 8. Integration
                                                                                                                                                                                                                                        • 8.1 Integration
                                                                                                                                                                                                                                          • 8.1.1 Fundamental Theorem of Calculus
                                                                                                                                                                                                                                            • 8.1.2 Integrating Powers of x
                                                                                                                                                                                                                                              • 8.1.3 Definite Integration
                                                                                                                                                                                                                                                • 8.1.4 Area Under a Curve
                                                                                                                                                                                                                                                  • 8.1.5 Area between a curve and a line
                                                                                                                                                                                                                                                  • 8.2 Further Integration
                                                                                                                                                                                                                                                    • 8.2.1 Integration as the limit of a sum
                                                                                                                                                                                                                                                      • 8.2.2 Integrating Other Functions (Trig, ln & e etc)
                                                                                                                                                                                                                                                        • 8.2.3 Reverse Chain Rule
                                                                                                                                                                                                                                                          • 8.2.4 f'(x)/f(x)
                                                                                                                                                                                                                                                            • 8.2.5 Substitution (Reverse Chain Rule)
                                                                                                                                                                                                                                                              • 8.2.6 Harder Substitution
                                                                                                                                                                                                                                                                • 8.2.7 Integrating with Trigonometric Identities
                                                                                                                                                                                                                                                                  • 8.2.8 Integration by Parts
                                                                                                                                                                                                                                                                    • 8.2.9 Integration using Partial Fractions
                                                                                                                                                                                                                                                                      • 8.2.10 Area between 2 curves
                                                                                                                                                                                                                                                                        • 8.2.11 Decision Making
                                                                                                                                                                                                                                                                        • 8.3 Differential Equations
                                                                                                                                                                                                                                                                          • 8.3.1 General Solutions
                                                                                                                                                                                                                                                                            • 8.3.2 Particular Solutions
                                                                                                                                                                                                                                                                              • 8.3.3 Separation of Variables
                                                                                                                                                                                                                                                                                • 8.3.4 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                  • 8.3.5 Solving & Interpreting Differential Equations
                                                                                                                                                                                                                                                                                • 9. Parametric Equations
                                                                                                                                                                                                                                                                                  • 9.1 Parametric Equations
                                                                                                                                                                                                                                                                                    • 9.1.1 Parametric Equations - Basics
                                                                                                                                                                                                                                                                                      • 9.1.2 Parametric Equations - Eliminating the Parameter
                                                                                                                                                                                                                                                                                        • 9.1.3 Parametric Equations - Sketching Graphs
                                                                                                                                                                                                                                                                                        • 9.2 Further Parametric Equations
                                                                                                                                                                                                                                                                                          • 9.2.1 Parametric Differentiation
                                                                                                                                                                                                                                                                                            • 9.2.2 Parametric Integration
                                                                                                                                                                                                                                                                                              • 9.2.3 Modelling with Parametric Equations
                                                                                                                                                                                                                                                                                            • 10. Numerical Methods
                                                                                                                                                                                                                                                                                              • 10.1 Solving Equations
                                                                                                                                                                                                                                                                                                • 10.1.1 Change of Sign
                                                                                                                                                                                                                                                                                                  • 10.1.2 Change of Sign Failure
                                                                                                                                                                                                                                                                                                    • 10.1.3 x = g(x) Iteration
                                                                                                                                                                                                                                                                                                      • 10.1.4 Newton-Raphson
                                                                                                                                                                                                                                                                                                        • 10.1.5 Trapezium Rule (Numerical Integration)
                                                                                                                                                                                                                                                                                                        • 10.2 Modelling involving Numerical Methods
                                                                                                                                                                                                                                                                                                          • 10.2.1 Numerical Methods in Context
                                                                                                                                                                                                                                                                                                        • 11. Vectors
                                                                                                                                                                                                                                                                                                          • 11.1 Vectors in 2 Dimensions
                                                                                                                                                                                                                                                                                                            • 11.1.1 Basic Vectors
                                                                                                                                                                                                                                                                                                              • 11.1.2 Magnitude & Direction
                                                                                                                                                                                                                                                                                                                • 11.1.3 Vector Addition
                                                                                                                                                                                                                                                                                                                  • 11.1.4 Position Vectors
                                                                                                                                                                                                                                                                                                                    • 11.1.5 Problem Solving using Vectors
                                                                                                                                                                                                                                                                                                                    • 11.2 Vectors in 3 Dimensions
                                                                                                                                                                                                                                                                                                                      • 11.2.1 Vectors in 3 Dimensions
                                                                                                                                                                                                                                                                                                                        • 11.2.2 Problem Solving using 3D Vectors


                                                                                                                                                                                                                                                                                                                        DOWNLOAD PDF

                                                                                                                                                                                                                                                                                                                      Author: Paul

                                                                                                                                                                                                                                                                                                                      Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.


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