1.01d Proof by contradiction

Prove by contradiction that \(\sqrt[3]{2}\) is an irrational number. [7 marks]

The three sides of a right-angled triangle have lengths \(a\), \(b\) and \(c\), where \(a, b, c \in \mathbb{Z}\) \includegraphics{figure_6}

1. State an example where \(a\), \(b\) and \(c\) are all even. [1 mark]
2. Prove that it is not possible for all of \(a\), \(b\) and \(c\) to be odd. [3 marks]

Assume that \(a\) and \(b\) are integers such that $$a^2 - 4b - 2 = 0$$

1. Prove that \(a\) is even. [2 marks]
2. Hence, prove that \(2b + 1\) is even and explain why this is a contradiction. [3 marks]
3. Explain what can be deduced about the solutions of the equation $$a^2 - 4b - 2 = 0$$ [1 mark]

Suppose \(x\) is an irrational number, and \(y\) is a rational number, so that \(y = \frac{m}{n}\), where \(m\) and \(n\) are integers and \(n \neq 0\). Prove by contradiction that \(x + y\) is not rational. [4]

Prove by contradiction that, for every real number \(x\) such that \(0 \leqslant x \leqslant \frac{\pi}{2}\), $$\sin x + \cos x \geqslant 1.$$ [4]

Prove by contradiction the following proposition: When \(x\) is real and positive, \(x + \frac{81}{x} \geq 18\). [4]

Prove by contradiction the following proposition. When \(x\) is real and positive, $$4x + \frac{9}{x} \geq 12.$$ The first line of the proof is given below. Assume that there is a positive and a real value of \(x\) such that $$4x + \frac{9}{x} < 12.$$ [3]

Prove by contradiction that there are no positive integers \(a\) and \(b\) with \(a\) odd such that $$a + 2b = \sqrt{8ab}$$ [4]

If \(a\) and \(b\) are odd integers such that 4 is a factor of \((a - b)\), prove by contradiction that 4 cannot be a factor of \((a + b)\). [3]

Prove by contradiction that \(\sqrt{7}\) is irrational. [5]

Shona makes the following claim. "\(n\) is an even positive integer greater than \(2 \Rightarrow 2^n - 1\) is not prime" Prove that Shona's claim is true. [4]

Prove by contradiction that there is no greatest even positive integer. [3]