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TMUA 2022 · Paper 2 · Question 9 of 20

TMUA 2022 Paper 2 Question 9

Logic and arguments — Quantified statements · finding all valid parameters. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Logic and argumentsQuantified statements · finding all valid parameters7 options
Consider the following statement:

(×) For all real numbers x, if x < k then x2 < k

What is the complete set of values of k for which (×) is true?

  1. Ano real numbers
  2. Bk > 0
  3. Ck < 1
  4. Dk ≤ 1
  5. E0 < k < 1
  6. F0 < k ≤ 1
  7. Gall real numbers
Show the answer and worked solution
answer · A
  1. Ano real numbers
  2. Bk > 0
  3. Ck < 1
  4. Dk ≤ 1
  5. E0 < k < 1
  6. F0 < k ≤ 1
  7. Gall real numbers
Whatever k is, choose x very large and negative. Then x < k holds, but x2 grows without bound and certainly exceeds k. So the implication fails for every k, and there is no value that makes (×) true.