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TMUA 2023 · Paper 2 · Question 11 of 20

TMUA 2023 Paper 2 Question 11

Logic and arguments — Statements equivalent to an implication. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Logic and argumentsStatements equivalent to an implication8 options
In this question, k is a positive integer. Consider the following theorem:

If 2k + 1 is a prime, then k is a power of 2.   (×)

Which of the following statements, taken individually, is/are equivalent to (×)?

I   If k is a power of 2, then 2k + 1 is prime.
II  2k + 1 is not prime only if k is not a power of 2.
III A sufficient condition for k to be a power of 2 is that 2k + 1 is prime.

  1. AI yes, II yes, III yes
  2. BI yes, II yes, III no
  3. CI yes, II no, III yes
  4. DI yes, II no, III no
  5. EI no, II yes, III yes
  6. FI no, II yes, III no
  7. GI no, II no, III yes
  8. HI no, II no, III no
Show the answer and worked solution
answer · G
  1. AI yes, II yes, III yes
  2. BI yes, II yes, III no
  3. CI yes, II no, III yes
  4. DI yes, II no, III no
  5. EI no, II yes, III yes
  6. FI no, II yes, III no
  7. GI no, II no, III yes
  8. HI no, II no, III no
Write (×) as P ⇒ Q, with P = "2k+1 is prime" and Q = "k is a power of 2". Statement I is the converse Q ⇒ P, which is not equivalent. Statement II says "¬ P only if ¬ Q", that is ¬ P ⇒ ¬ Q — the inverse, equivalent to the converse and so not to (×). Statement III says P is sufficient for Q, which is exactly P ⇒ Q. Only III.