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Suppose S is a finite set with 7 elements. How many elements are there in the largest equivalence relation on S?

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Suppose S is a finite set with 7 elements. How many elements are there in the largest equivalence relation on S?

(a) 56

(b) 78

(c) 49

(d) 100

This question was addressed to me by my college professor while I was bunking the class.

The query is from Number of Relations topic in portion Relations of Discrete Mathematics

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The correct choice is (c) 49

Easiest explanation: Let R is an equivalence relation on the set S and so it satisfies the reflexive, symmetric and transitive property. The largest equivalence relation means it should contain the largest number of ordered pairs. Since we can have n^2 ordered pairs in R x R where n belongs to S and all these ordered pairs are present in this relation; its the largest equivalence relation.So there are n^2 elements i.e 7^2 = 49 elements in the largest equivalence relation.

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