1.1 Proof by Contradiction

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A cybersecurity expert is verifying a protocol where a security key $k\,$ is generated as the quotient $k=μσ\displaystyle k = \frac{\mu}{\sigma}$ , where $\mu\,$ is a non-zero rational coefficient and $\sigma\,$ is an irrational physical constant. To establish the security of the protocol, the expert wishes to use proof by contradiction to prove the statement:

'The quotient of a non-zero rational number and an irrational number is always irrational.'

Identify the correct starting assumption for the expert's proof.

For all non-zero rational $\mu\,$ and irrational $σ\sigma$ , the quotient $μσ\displaystyle \frac{\mu}{\sigma}$ is rational.
There exists a non-zero rational $\mu\,$ and an irrational $\sigma\,$ such that the quotient $μσ\displaystyle \frac{\mu}{\sigma}$ is rational.
There exists a non-zero rational $\mu\,$ and an irrational $\sigma\,$ such that the quotient $μσ\displaystyle \frac{\mu}{\sigma}$ is irrational.
For all non-zero rational $\mu\,$ and irrational $σ\sigma$ , the quotient $μσ\displaystyle \frac{\mu}{\sigma}$ is irrational.

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1.1 Proof by Contradiction Questions

Practise Edexcel A Level Maths 1.1 Proof by Contradiction with exam-style questions for A Level Maths. 100 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.