Let S be a set of n elements say (1, 2, 3,... n).
Now the smallest equivalence relation on S must contain all the reflexive elements ((1, 1), (2, 2). (3, 3),..., (n, n)} and its cardinality is therefore n.
The largest equivalence relation on S is Sx S, which has cardinality of nxn = r².
The largest and smallest equivalence relations on S have cardinalities of n² and n respectively.