Proof

Proof by Contradiction

Use congruence in polygons as a method of proof

An equation has an equals sign, "=". You can solve it to find the value of the letter. An identity is similar, but is true for all values of x and uses "≡".

To show a statement is an identity, expand and simplify the expressions on one or both sides of the identity sign, until the two expressions are the same.

Consecutive integers can be written as $n, (n + 1), (n + 2), (n + 3), ...$ where $n \in ℤ$

Useful Expressions

The $m^{th}$ even number is given by $2 m$ .

The $n^{th}$ odd number is given by $2 n - 1$ .

Circle Theorems Proof

Radius and Tangent proof
Right angle in a semi-circle
angles at circumference\centre
Angles in the same segment
Cyclic quadrilaterals
Alternate segment theorem: Practice Questions; Textbook Exercise

Quadratic formula proof: $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$; - the algebra; Practice Questions; Textbook Exercise

Trigonometry:
Area of a triangle: $A = \frac{1}{2} a b sin C$; Proof; Practice Questions; Textbook Exercise

Algebraic Proof Video: Practice Questions; Textbook Exercise
Geometric Proof Video: (Care of corbettmaths.com)
Proof: (Care of www.mathsgenie.co.uk)
Proof Worksheets: (Care of mmerevise.co.uk)
Identities:
Equating coefficients: (Care of corbettmaths.com)
A level: AQA Proof; Edexcel Proof; OCR Proof; (Care of www.physicsandmathstutor.com)

Various Differentiation Proofs

Proof by Contradiction

What is Proof by Contradiction?

To prove a mathematical statement by contradiction, we begin by assuming the opposite of what we want to prove, i.e., we assume the statement is false. Through a series of logical steps and established mathematical principles, we then demonstrate that this assumption leads to a contradiction. This is either by conflicting with known facts or by producing an internal inconsistency. This contradiction implies that our initial assumption must be false, and therefore, the original statement must be true.

When we use proof by contradiction, we start by pretending the statement we want to prove is actually false. This is called the 'negation.' Then, using logic and what we already know in maths, we follow that assumption to see where it leads. If it ends up clashing with a known fact or creates something impossible, we have found a contradiction. That tells us our original assumption (the negation) must be wrong—so the statement we wanted to prove must be true.

Example: Step-by-Step Proof that $\sqrt{2}$ is Irrational

Step 1:
Assume

\sqrt{2}

is rational (this is the negation).
So,

\sqrt{2} = \frac{a}{b}

where

a

and

b

are whole numbers with no common factors.

Step 2:
Square both sides:

2 = \frac{a^{2}}{b^{2}} \to 2 b^{2} = a^{2}

Step 3:
Since

a^{2}

is even,

a

must be even.
So, let

a = 2 k

Step 4:
Substituting back,

a^{2} = 4 k^{2} \to 2 b^{2} = 4 k^{2} \to b^{2} = 2 k^{2}

a

is also even.

Step 5:
Both a and b are even, which contradicts our assumption that they have no common factors.

Step 6:
Conclusion.
Therefore,

\sqrt{2}

must be irrational.

Quick Quiz: Test Your Understanding

1. What does it mean if a number is irrational?

It can be written as a fraction It cannot be written as a fraction It is divisible by 2

2. Why is assuming $\sqrt{2}$ is rational useful in this proof?

It helps us find its exact value It leads to a contradiction It proves that

\sqrt{2}

is even

Solution Bank Yr 2 Pure ex 1A
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Questions:

1. Prove that for all positive values of $x$ and $y$ $\sqrt{x y} \leq \frac{x + y}{2}$
  (2)
  
  Solution:
  
  $x y \leq \frac{{(x + y)}^{2}}{4}$ $4 x y \leq {(x + y)}^{2}$ $4 x y \leq x^{2} + 2 x y + y^{2}$ $0 \leq x^{2} - 2 x y + y^{2}$ $0 \leq {(x - y)}^{2}$
  
  -oOo-
2. Prove by counter example that the inequality is not true when $x$ and $y$ are both negative.
  (1)
  
  Solution:
  
  Choose any two negative numbers, e.g., $x = -1$ and $y = -3$ then LHS is $\sqrt{-1 \times -3} = \sqrt{3}$ and the RHS is $\frac{-1 + -3}{2} = -2$ . Therefore LHS ≰ RHS.
  
  -oOo-

Proof by Exhaustion

Questions

Prove by exhaustion that the sum of two even positive integers less than 10 is also even. (3 marks)
Show that $n^{2} + 2$ is not a multiple of 4, where $n$ is an integer, $2 \leq n \leq 7$ .
If $p$ is a prime number such that $5 \leq p \leq 23$ , then prove by exhaustion that $p^{2} - 1$ is a multiple of 12.
Proof that a multiple of seven and $n^{3} + 7$ differ by one. For all integers in the domain $1 \leq n \leq 4$ . (3 marks)
Use proof by exhaustion to show that for $n \in ℕ, n \leq 4, {(n + 1)}^{3} < 3^{n}$ . (2 marks)
Prove that no square number ends in a 7. (3 marks)
Show that $n^{2} - 1$ is a multiple of 3 if $n$ is not a multiple of 3.

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Binomial Expansion

Prove

{(1 + x)}^{n} = 1 + n x + \frac{n (n - 1) x^{2}}{2!} + \frac{n (n - 1) (n - 2) x^{3}}{3!} + ...

Solution:

Given

{(a + b)}^{n} = Σ_{k = 0}^{n} (\begin{matrix} n \\ k \end{matrix}) a^{n - k} b^{k}

then

{(1 + x)}^{n} = Σ_{k = 0}^{n} (\begin{matrix} n \\ k \end{matrix}) x^{k}

Expanding

{(1 + x)}^{n} = \frac{n!}{n!} x^{0} + \frac{n!}{1! (n - 1)!} x^{1} + \frac{n!}{2! (n - 2)!} x^{2} + \frac{n!}{3! (n - 3)!} x^{3} + ...

Simplifying

{(1 + x)}^{n} = 1 + n x + \frac{n (n - 1) x^{2}}{2!} + \frac{n (n - 1) (n - 2) x^{3}}{3!} + ...

As required.

Disproof By Counter Example

A mathematical statement is not true by using a counter-example. A counterexample is an example that doesn't work for the statement. There is no need to give more than one counter-example, as one is sufficient to disprove a statement.

For example, to prove that " $4 n + 4$ is a multiple of eight for all $n$ " is not true by the counter-example, $n = 2$ , as $4 \times 2 + 4 = 12$ and $12$ is not a multiple of eight.

Questions:

Disprove the statement, "If $a > b$ , then $a^{2} > b^{2}$ for all values of $a$ and $b$ ."

-oOo-

Proof by Exhaustion

Disproof by Counter Example

Proof by Contradiction

Useful Expressions

Circle Theorems Proof

Various Differentiation Proofs

Proof by Contradiction

What is Proof by Contradiction?

Example: Step-by-Step Proof that $\sqrt{2}$ is Irrational

Quick Quiz: Test Your Understanding

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Questions:

Proof by Exhaustion

Questions

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Binomial Expansion

Solution:

Disproof By Counter Example

Questions:

Disproof by Counter Example
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Proof by Counter Example
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Proof By Counter-Example
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Proof by Exhaustion

Disproof by Counter Example

Proof by Contradiction

Useful Expressions

Circle Theorems Proof

Various Differentiation Proofs

Proof by Contradiction

What is Proof by Contradiction?

Example: Step-by-Step Proof that 2 is Irrational

Quick Quiz: Test Your Understanding

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Questions:

Proof by Exhaustion

Questions

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Binomial Expansion

Solution:

Disproof By Counter Example

Questions:

Disproof by Counter Example (Care of www.savemyexams.com)

Proof by Counter Example (Care of www.advancedhighermaths.co.uk)

Proof By Counter-Example (Care of www.advancedhighermaths.co.uk/proof-counter-example/)

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Example: Step-by-Step Proof that $\sqrt{2}$ is Irrational

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Disproof by Counter Example
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Proof by Counter Example
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Proof By Counter-Example
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