1.3.7

Proof by Contradiction (A2 Only)

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What is Proof by Contradiction?

Proof by contradiction is when we assume that the conjecture is false and use a series of logical steps to obtain a statement that is mathematically impossible. This proves that the original conjecture is always true.

How do you prove by contradiction?

How do you prove by contradiction?

  • To prove a conjecture by contradiction, assume that the conjecture is always false.
  • Next, we use known theorems to make logical steps from the assumption.
  • You will reach a point where, by inspection, the logical step either:
    • Contradicts the starting assumption.
    • Violates a known theorem.
  • This proves that the original conjecture is always true.
Example

Example

  • Prove that there are no integers xx and yy that satisfy:
    • x2−y2=10x^2 - y^2 = 10
Assumption

Assumption

  • If the conjecture is false, then that means the following equation is true:
    • x2−y2=10x^2-y^2 = 10
  • Where x,y∈x,y \in ℤ (are integers).
Logical steps

Logical steps

  • Looking at the left hand side, we see that it is the difference of two squares:
    • x2−y2≡(x+y)(x−y)=10x^2-y^2 \equiv (x+y)(x-y) = 10
  • We don't have any more information about xx and yy except that they are integers, so we need to look at what happens for the different cases of xx and yy.
Both $$x$$ and $$y$$ are odd

Both xx and yy are odd

  • It is a known theorem that odd + odd = even and odd − odd = even.
    • So x+y=2mx + y = 2m and x−y=2nx-y = 2n, where mm and nn are integers.
    • This means (x+y)(x−y)=4mn(x+y)(x-y) = 4mn.
  • By inspection, we see that the left-hand side of our initial assumption is a multiple of 4.
  • This is a contradiction as the right-hand side is equal to 10, which is not a multiple of 4.
Diagram
Both $$x$$ and $$y$$ are even

Both xx and yy are even

  • It is a known theorem that even + even = even and even − even = even.
    • So x+y=2px + y = 2p and x−y=2qx-y = 2q, where pp and qq are integers.
    • This means (x+y)(x−y)=4pq(x+y)(x-y) = 4pq.
  • By inspection, we see that the left-hand side is a multiple of 4.
  • This is a contradiction, as 10 is not a multiple of 4.
Diagram
$$x$$ is even, $$y$$ is odd

xx is even, yy is odd

  • It is a known theorem that even + odd = odd and even − odd = odd.
    • So x+y=2k+1x+y=2k+1 and x−y=2l+1x-y = 2l+1, where kk and ll are integers.
    • This means (x+y)(x−y)=(2k+1)(2l+1)=4kl+2l+2k+1(x+y)(x-y) = (2k+1)(2l+1) = 4kl + 2l + 2k + 1.
  • By inspection, we see that the left-hand side is equal to even + even + even + odd, which is an odd number.
  • This is a contradiction as 10 is an even number.
    • The same applies for odd xx and even yy.
Diagram
Final statement

Final statement

  • There are no integers xx and yy that satisfy:
    • x2−y2=10x^2-y^2 = 10
Jump to other topics
1

Proof

1.1

Types of Numbers

1.1.1Whole Numbers1.1.2The Four Rules for Positive Numbers1.1.3Negative Numbers1.1.4Integers1.1.5The Four Rules for Negative Numbers1.1.6Real Numbers1.1.7Rational vs Irrational
1.2

Notation

1.2.1Sets1.2.2Subsets1.2.3Venn Diagrams1.2.4Sets Numbers1.2.5Notation in Mathematical Arguments
1.3

Proof

1.3.1Mathematical Proof1.3.2Proof by Deduction1.3.3Proving an Identity1.3.4Polynomial Identities1.3.5Proof by Exhaustion1.3.6Disproof by Counter Example1.3.7Proof by Contradiction (A2 Only)
2

Algebra & Functions

2.1

Powers & Roots

2.1.1Exponential Rules2.1.2Rational Exponents2.1.3Surds2.1.4Simplify Surds2.1.5Rationalise Denominator
2.2

Quadratic Equations

2.2.1Quadratic +/-2.2.2Factorise Quadratics2.2.3Quadratic Formula2.2.4Complete the Square2.2.5Solve Quadratics Graphically2.2.6Simultaneous Equations2.2.7Simultaneous Quadratic Equations2.2.8Simultaneous Equations Graphs2.2.9Simultaneous Equations Word Problems2.2.10Quadratic Equations of Powers
2.3

Inequalities

2.3.1Solving Inequalities2.3.2Solving Double Inequalities2.3.3Solving Quadratic Inequalities2.3.4Representing Inequalities2.3.5Graphing Inequalities2.3.6Set Notation2.3.7Set Notation & Inequalities
2.4

Polynomials

2.4.1Polynomials2.4.2Adding & Subtracting Polynomials2.4.3Remainder Theorem2.4.4Algebraic Long Division2.4.5Rewriting Rational Expressions
2.5

Graphs

2.5.1Graphs of Polynomials2.5.2Simultaneous Equations Graphs2.5.3Simultaneous Quadratic Equations2.5.4Graphs of Proportion2.5.5Graphs of Common Functions 12.5.6Graphs of Common Functions 22.5.7Graphs of Common Functions 32.5.8Proportion2.5.9Direct Proportion2.5.10Inverse Proportion
2.6

Functions

2.6.1Properties of Functions2.6.2Inverse Functions2.6.3Composite Functions
2.7

Transformation of Graphs

2.7.1Reflections2.7.2Rotation2.7.3Translation2.7.4Stretches & Compressions2.7.5Effects of Transformations2.7.6Combinations of Transformations
2.8

Partial Fractions (A2 Only)

2.8.1Partial Fractions2.8.2Partial Fractions with Three Linear Factors2.8.3Partial Fractions with Three Linear Factors 22.8.4Partial Fractions with Repeated Factors2.8.5Partial Fractions - Practice
3

Coordinate Geometry

3.1

Straight Lines

3.1.1y = mx +c3.1.2Gradient & Y-intercept3.1.3Distance-Time Graph3.1.4Velocity-Time Graph3.1.5Conversion Graph
3.2

Circles

3.2.1Equations of Circles centred at Origin3.2.2Finding the Centre & Radius3.2.3Equation of a Tangent3.2.4Circle Theorems - Perpendicular Bisector3.2.5Circle Theorems - Angle at the Centre3.2.6Circle Theorems - Angle at a Semi-Circle3.2.7Equation of a Perpendicular Bisector3.2.8Equation of a Circumcircle3.2.9Circumcircle of a Right-angled Triangle
3.3

Parametric Equations (A2 only)

3.3.1Parametric Equations3.3.2Modelling with Parametric Equations
4

Sequences & Series

4.1

Binomial Expansion

4.1.1Pascal's Triangle4.1.2Factorials4.1.3Binomial Expansion4.1.4Binomial Expansion & Rational Powers (A2 only)4.1.5Binomial Expansion & Rational Powers 2 (A2 only)
4.2

Sequences (A2 only)

4.2.1Nth term Rules4.2.2Recurring Rules4.2.3Types of Sequences4.2.4Arithmetic Sequences4.2.5Geometric Sequences4.2.6Sigma Notation4.2.7Arithmetic Series4.2.8Geometric Series4.2.9Sum to Infinity of a Geometric Series4.2.10Modelling with Sequences & Series
5

Trigonometry

5.1

Using Angles

5.1.1Radians5.1.2Arc length5.1.3Area of Sectors5.1.4Area of Segments
5.2

Trigonometric Functions

5.2.1Unit Circle5.2.2Properties of Trig Functions5.2.3Trigonometry5.2.4Exact values5.2.5Graphs of Trigonometric Functions5.2.6Small Angle Approximation
5.3

Triangle Rules

5.3.1Sine Rules5.3.2Cosine Rules5.3.3Area of a Triangle
5.4

Trigonometric Identities

5.4.1Secant, Cosecant & Cotangent5.4.2Pythagorean Identities5.4.3Angle Addition Formulae5.4.4Manipulating the Angle Addition Formulae5.4.5Double Angle Formula
6

Exponentials & Logarithms

6.1

Exponentials & Logarithms

6.1.1Exponential Function6.1.2Logarithms6.1.3Properties of Logarithms6.1.4Natural Logarithm6.1.5Graphs of Exponents & Logarithms
6.2

Exponential Growth Decay

6.2.1Population Growth6.2.2Depreciation6.2.3Graphing Linear and Exponential Growth6.2.4Interpreting Exponential Growth and Decay Graphs6.2.5Compound Growth & Decay6.2.6Compund Interest
7

Differentiation

7.1

Derivatives

7.1.1Gradient of a Curve7.1.2Differentiation from First Principles7.1.3Differentiation from First Principles - Practice7.1.4Differentiation7.1.5Finding derivatives7.1.6Second Order Derivatives
7.2

Graphs & Differentiation

7.2.1Stationary Points7.2.2Maxima & Minima7.2.3Modelling with Differentiation7.2.4Using Second Derivatives (A2 only)7.2.5Rate of Change (A2 only)
7.3

Differentiation With Trigonometry and Exponentials

7.3.1Differentiation with Sin & Cos7.3.2Diff. of Exponential Functions7.3.3Differentiation of Trig. Functions 1 (A2 only)7.3.4Differentiation of Trig. Functions 2 (A2 only)
7.4

Rules of Differetiation (A2 only)

7.4.1Chain Rule7.4.2Product Rule7.4.3Quotient Rule
7.5

Parametric & Implicit Differentiation

7.5.1Parametric Differentiation7.5.2Implicit Differentiation
8

Integration

8.1

Integration

8.1.1Integration - Basics8.1.2Integration - Definite Integrals8.1.3Finding Functions8.1.4Area under curves8.1.5Area under x-axis8.1.6Area between curves and lines
8.2

Integration (A2 Only)

8.2.1Integration - Standard Functions8.2.2Integration - Reverse Chain Rule8.2.3Integration by Substitution8.2.4Integration by Parts8.2.5Integration by Parts - Practice8.2.6Integration by Partial Fraction8.2.7Modelling with Differential Equations
9

Numerical Methods

9.1

Finding Solutions

9.1.1Number Iterations9.1.2Iterations to 1 Decimal Places9.1.3Iteration Formula9.1.4Locating Roots9.1.5Newton Raphson Method
9.2

Finding the Area

9.2.1Trapezium Rule
10

Vectors

10.1

2D Vectors

10.1.1Vectors - Magnitude & Direction10.1.2Unit Vectors10.1.3Vectors - Algebraic Operations10.1.4Position Vectors10.1.5Vector Proofs10.1.6Vectors Word Questions
10.2

3D Vectors

10.2.13D Vector Magnitude10.2.23D Vectors - Algebraic Operations10.2.3Position Vectors 3D10.2.43D Vectors Word Questions
10.3

Vector Proofs

10.3.1Coming Soon

Practice questions on Proof by Contradiction (A2 Only)

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  1. 1
    To prove the following statement by contradiction, which of the following should you take as your initial assumption? <br>All multiples of 6 are even.Multiple choice
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Disproof by Counter Example

Exponential Rules