TTMUA Lab
TMUA 2022 · Paper 2 · Question 10 of 20

TMUA 2022 Paper 2 Question 10

Logic and arguments — Order of quantifiers. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Logic and argumentsOrder of quantifiers8 options
Which of the following statements is/are true?

I   For all real numbers x and for all positive integers n, x < n
II  For all real numbers x, there exists a positive integer n such that x < n
III There exists a real number x such that for all positive integers n, x < n

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · G
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
I is false: take x = 100 and n = 1. II is true — given any real x there is always a positive integer beyond it. III is true as well, and needs only one witness: x = 0 is less than every positive integer. The order of the quantifiers is what separates II from III.