Question

The number of equivalence relations of the set {3, 6, 9, 12, 18} is ______

(a) 4

(b) 2^5

(c) 22

(d) 90

This question was posed to me by my school principal while I was bunking the class.

This intriguing question originated from Closure on Relations in division Relations of Discrete Mathematics

Answer 1

The correct answer is (a) 4

For explanation I would say: Number of equivalence Relations are given by BELL number. The nth of these numbers i.e, Bn counts the number of different ways to partition a set that has exactly n elements, or equivalently, the number of equivalence relations on it. Let’s say, 1 -> Equivalence relation with 1 element; 1 2 -> Equivalence relation with 2 element; 2 3 5 -> Equivalence relation with 3 element; 5 7 10 15 -> Equivalence relation with 4 element. Hence, the answer is 4.