1.3.7
Proof by Contradiction (A2 Only)
What is Proof by Contradiction?
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 and
that satisfy:
Assumption
Assumption
- If the conjecture is false, then that means the following equation is true:
- Where ℤ (are integers).
Logical steps
Logical steps
- Looking at the left hand side, we see that it is the difference of two squares:
- We don't have any more information about and except that they are integers, so we need to look at what happens for the different cases of and .
Both and
are odd
Both and are odd
- It is a known theorem that odd + odd = even and odd − odd = even.
- So and , where and are integers.
- This means .
- 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.
Both and
are even
Both and are even
- It is a known theorem that even + even = even and even − even = even.
- So and , where and are integers.
- This means .
- 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.
is even,
is odd
is even, is odd
- It is a known theorem that even + odd = odd and even − odd = odd.
- So and , where and are integers.
- This means .
- 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 and even .
Final statement
Final statement
- There are no integers and
that satisfy:
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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