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TMUA 2020 · Paper 2 · Question 20 of 20

TMUA 2020 Paper 2 Question 20

Logic and arguments — Implications · spotting equivalent statements. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2Logic and argumentsImplications · spotting equivalent statements6 options
x is a real number and f is a function.

Given that exactly one of the following statements is true, which one is it?

  1. Ax ≥ 0 only if f(x) < 0
  2. Bx < 0 if f(x) ≥ 0
  3. Cx ≥ 0 only if f(x) ≥ 0
  4. Df(x) < 0 if x < 0
  5. Ef(x) ≥ 0 only if x ≥ 0
  6. Ff(x) ≥ 0 if and only if x < 0
Show the answer and worked solution
answer · C
  1. Ax ≥ 0 only if f(x) < 0
  2. Bx < 0 if f(x) ≥ 0
  3. Cx ≥ 0 only if f(x) ≥ 0
  4. Df(x) < 0 if x < 0
  5. Ef(x) ≥ 0 only if x ≥ 0
  6. Ff(x) ≥ 0 if and only if x < 0
Write P for "x ≥ 0" and Q for "f(x) ≥ 0", so that ¬ P is "x < 0" and ¬ Q is "f(x) < 0". The six options become A: P ⇒ ¬ Q; B: Q ⇒ ¬ P; C: P ⇒ Q; D: ¬ P ⇒ ¬ Q; E: Q ⇒ P; F: Q ⇔ ¬ P. Now look for contrapositives: B is the contrapositive of A, so A and B are always both true or both false, and neither can be the unique true one; likewise D is the contrapositive of E, ruling out both. That leaves C and F, and F is the conjunction of Q ⇒ ¬ P (which is B) with ¬ P ⇒ Q, so F being true would drag B along with it. Only C can stand alone, and it does: taking f(x) = 1 for every x makes C true and all the others false.