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Let A = 1, 2, 3, 4 . The number of equivalence relations on the set A that do not contain the ordered pair (1,2) is :

Options

  1. A15
  2. B5
  3. C14
  4. D10

Correct answer

D. 10

Step-by-step solution

The number of equivalence relations on a set of n elements is given by the n -th Bell number B_n . For n=4 , the total number of equivalence relations is B₄ = 15 . Let us find the number of equivalence relations that contain the ordered pair (1,2) . Since an equivalence relation is symmetric and transitive, containing (1,2) implies that 1 and 2 must belong to the same equivalence class. We can treat 1,2 as a single element. The problem then reduces to finding the number of equivalence relations on a set of 3 elemen

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