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Let A be a set containing exactly 6 elements. If an equivalence relation R on the set A contains exactly 14 ordered pairs, then the number of equivalence classes of R is:

Options

  1. A3
  2. B2
  3. C4
  4. D6

Correct answer

A. 3

Step-by-step solution

Let the equivalence relation R partition the set A into k disjoint equivalence classes of sizes a₁, a₂, , a_k . The total number of elements in the set A is the sum of the sizes of these classes: a₁ + a₂ + + a_k = 6 The number of ordered pairs in an equivalence relation is the sum of the squares of the sizes of its equivalence classes. Thus, we are given: a₁^2 + a₂^2 + + a_k^2 = 14 We must find a partition of the integer 6 such that the sum of the squares of its parts is 14 . Let us test the possible partitions of

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