IAL Edexcel P4 Proof by Contradiction

IAL P4 Proof by Contradiction

Use proof by contradiction to prove true statements

Integers

An integer (Z) is a whole number, definined as a number that can be written without a fractional component.

Rational and Irrational numbers

A rational number (Q) can be written as

 $\frac{a}{b}$  where a and b are integers

An irrational number (Q') cannot!

Prime numbers

A prime number can only be divided by itself and 1

E.g. 3 = 3x1 only

Multiple Choice

Which symbol represents natural numbers?

• N

• Z

• Q

• R

Multiple Choice

Which symbol represents integers?

• N

• Z

• Q

• R

Multiple Choice

Which symbol represents rational numbers?

• N

• Z

• Q

• R

Proof by Contradiction

Step 1: Begin by assuming the original statement is FALSE

(Negation: the contradiction or denial of something)

Multiple Choice

Select the negation to:

There is NO greatest odd number

• There is a greatest odd number

• There is NO greatest even number

Multiple Choice

Select the negation to:

 $\sqrt{2}$  is an irrational number

•  $2^2$  is an irrational number

•  $\sqrt{2}$  is a rational number
  

Multiple Choice

Select the negation to:

There is an infinite number of prime numbers

• There is an infinite number of non prime numbers

• There is a finite number of primes

Proof by Contradiction

Step 2: Use logical steps to attempt to prove the false statement

Multiple Choice

Which expression always represents an even number?

• n

• n+2

• 2n

• n²

Multiple Choice

Which expression always represents an odd number?

• 2n+1

• n+2

• 2n

• n²+1

Multiple Choice

Which expression always represents a rational number?

• $\frac{a}{b}$

• $\frac{a}{\sqrt{b}}$

• $\sqrt{a}$
  

• $\frac{\sqrt{a}}{b}$
  

Proof by Contradiction

Step 3: Finish your proof by conlcuding that the original statement MUST be true

Exercise 1A

Work through questions 1-10 on pages 4 and 5