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TMUA 2022 · Paper 2 · Question 6 of 20

TMUA 2022 Paper 2 Question 6

Logic and arguments — Necessary and sufficient · medians. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Logic and argumentsNecessary and sufficient · medians4 options
A list consists of n integers.

Consider the following statements:

P:   n is odd.
Q:   The median of the list is one of the numbers in the list.

Which one of the following is true?

  1. AP is necessary and sufficient for Q.
  2. BP is necessary but not sufficient for Q.
  3. CP is sufficient but not necessary for Q.
  4. DP is not necessary and not sufficient for Q.
Show the answer and worked solution
answer · C
  1. AP is necessary and sufficient for Q.
  2. BP is necessary but not sufficient for Q.
  3. CP is sufficient but not necessary for Q.
  4. DP is not necessary and not sufficient for Q.
If n is odd the median is the single middle entry, which is certainly a member of the list — so P is sufficient. It is not necessary: the list 1,  2,  2,  3 has an even length, yet its median is 2+22 = 2, which does appear in the list.