Edexcel IAL - Pure Maths 4- 1.1 Proof by Contradiction- Study notes  - New syllabus

Proof by Contradiction

Proof by contradiction is a method in which a statement is proved true by assuming that it is false and then showing that this leads to an impossibility.

If assuming the opposite of a statement leads to a contradiction, then the original statement must be true.

Logical Structure

To prove a statement \( P \):

• Assume \( P \) is false
• Use logical steps to derive a contradiction
• Conclude that the assumption is impossible
• Therefore, \( P \) must be true

Proof that \( \sqrt{2} \) is Irrational

We prove that \( \sqrt{2} \) cannot be written as a fraction of two integers.

Assume that \( \sqrt{2} \) is rational.

Then:

\( \sqrt{2} = \dfrac{p}{q} \)

where \( p \) and \( q \) are integers with no common factors and \( q \ne 0 \).

Squaring both sides:

\( 2 = \dfrac{p^2}{q^2} \Rightarrow p^2 = 2q^2 \)

This shows that \( p^2 \) is even, so \( p \) is even.

Let \( p = 2k \) for some integer \( k \).

Then:

\( p^2 = 4k^2 = 2q^2 \Rightarrow q^2 = 2k^2 \)

This shows that \( q \) is also even.

So both \( p \) and \( q \) are even, which contradicts the assumption that they have no common factor.

Therefore, \( \sqrt{2} \) is irrational.

Proof that There Are Infinitely Many Prime Numbers

We prove this using contradiction.

Assume that there are only finitely many primes.

Let them be:

\( p_1, p_2, p_3, \dots, p_n \)

Consider the number:

\( N = p_1p_2p_3\cdots p_n + 1 \)

When \( N \) is divided by any prime \( p_i \), it leaves remainder 1.

So \( N \) is not divisible by any of the listed primes.

This means \( N \) is either prime itself or has a prime factor not in the list.

This contradicts the assumption that all primes were listed.

Therefore, there are infinitely many primes.

Application to an Unfamiliar Result

Statement: The number \( 3 + \sqrt{5} \) is irrational.

Assume that \( 3 + \sqrt{5} \) is rational.

Then:

\( 3 + \sqrt{5} = r \)

for some rational number \( r \).

So:

\( \sqrt{5} = r – 3 \)

The right-hand side is rational, since rational numbers are closed under subtraction.

This implies that \( \sqrt{5} \) is rational, which is false.

This is a contradiction.

Therefore, \( 3 + \sqrt{5} \) is irrational.

Common Mistakes to Avoid

• Forgetting to state the initial assumption clearly
• Not identifying the contradiction
• Using circular reasoning
• Stopping before concluding the result