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Question

What is the minimum number of elements an equivalence relation defined on the set A {1,2,3} would have?


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Solution

For a relation to be equivalence, it should be symmetric, reflexive and transitive. We saw that an empty relation is symmetric and transitive because it does not violate the conditions. To make the empty relation reflexive, we have to add elements of the form (a,a) for all 'a' from the given set. They are (a,a), (b,b) and (c,c). So by adding these elements the empty set becomes the set {(a,a),(b,b), (c,c) }. This set is symmetric and transitive also. So the minimum number of elements in an equivalence relation defined on the given set is 3.

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