Existence of greatest/smallest element

A question is this type if and only if it asks to prove by contradiction that there is no greatest/smallest element in a set (e.g., no greatest odd integer, no smallest value in an interval).

5 questions · Moderate -0.7

1.01d Proof by contradiction
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Edexcel P4 2021 January Q3
2 marks Moderate -0.8
3. Prove by contradiction that there is no greatest odd integer.
OCR H240/01 2020 November Q4
3 marks Moderate -0.8
4 Prove by contradiction that there is no greatest multiple of 5 .
OCR H240/01 Q6
3 marks Moderate -0.5
6 Prove by contradiction that there is no greatest even positive integer.
AQA Paper 2 2024 June Q11
6 marks Moderate -0.8
  1. A student states that 3 is the smallest value of \(k\) in the interval \(3 < k < 4\) Explain the error in the student's statement. [1 mark]
  2. The student's teacher says there is no smallest value of \(k\) in the interval \(3 < k < 4\) The teacher gives the following correct proof: Step 1: Assume there is a smallest number in the interval \(3 < k < 4\) and let this smallest number be \(x\) Step 2: let \(y = \frac{3 + x}{2}\) Step 3: \(3 < y < x\) which is a contradiction. Step 4: Therefore, there is no smallest number in interval \(3 < k < 4\)
    1. Explain the contradiction stated in Step 3 [1 mark]
    2. Prove that there is no largest value of \(k\) in the interval \(3 < k < 4\) [4 marks]
OCR H240/01 2017 Specimen Q6
3 marks Moderate -0.5
Prove by contradiction that there is no greatest even positive integer. [3]