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Daniel J Smith Custom TMUA Set A Paper 2 · Q6

Paper 2Logic & proof

A statement SS is given:

For all real numbers xx, if x2≥1x^2\geq1 then x4≥x2x^4\geq x^2

A student wishes to prove SS by contradiction and writes the following argument:

Assume that there exists a real number xx such that x2≥1x^2\geq1 and x4<x2x^4<x^2

  • I. Then x4−x2<0x^4-x^2<0
  • II. So x2(x2−1)<0x^2(x^2-1)<0
  • III. So x2−1<0x^2-1<0
  • IV. So x2<1x^2<1, which contradicts x2≥1x^2\geq1

Therefore the original statement SS is concluded to be true.

Which of the following best describes this argument?

AIt is completely correct.
BIt is incorrect, but it would be correct if written in the reverse order.
CIt is incorrect, and the student has actually shown that SS is false.
DIt is incorrect because line II does not follow from line I.
EIt is incorrect because line III does not follow from line II.
FIt is incorrect because the student has not really used proof by contradiction.

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