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Let R be an equivalence relation on the set A = 1, 2, 3, 4, 5 such that 1 R 2 , 3 R 4 , and 2 R 3 . The minimum possible number of elements in R is

Options

  1. A8
  2. B13
  3. C9
  4. D7

Correct answer

C. 9

Step-by-step solution

An equivalence relation partitions the set A into disjoint equivalence classes. The number of elements in R is the sum of the squares of the sizes of these equivalence classes. Given 1 R 2 and 3 R 4 , the elements 1 and 2 must belong to the same equivalence class, and 3 and 4 must belong to the same equivalence class. The condition 2 R 3 ensures that the equivalence class containing 1, 2 is disjoint from the equivalence class containing 3, 4 . To minimize the total number of elements in R , the remaining element 5

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