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(A) The number of equivalence relations on a set with $n$ elements is equal to the number of partitions of the set,which is given by the Bell number $

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$14 \leq N \leq 20$

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(A) The number of equivalence relations on a set with $n$ elements is equal to the number of partitions of the set,which is given by the Bell number $B_n$. For a set $A$ with $n = 4$ elements,the number of equivalence relations is the Bell number $B_4$. The Bell numbers are defined by the recurrence relation $B_{n+1} = \sum_{k=0}^{n} \binom{n}{k} B_k$. Calculating the first few Bell numbers: $B_0 = 1$ $B_1 = 1$ $B_2 = \binom{1}{0}B_0 + \binom{1}{1}B_1 = 1(1) + 1(1) = 2$ $B_3 = \binom{2}{0}B_0 + \binom{2}{1}B_1 + \binom{2}{2}B_2 = 1(1) + 2(1) + 1(2) = 5$ $B_4 = \binom{3}{0}B_0 + \binom{3}{1}B_1 + \binom{3}{2}B_2 + \binom{3}{3}B_3 = 1(1) + 3(1) + 3(2) + 1(5) = 1 + 3 + 6 + 5 = 15$. Thus,$N = 15$. Checking the options: $14 \leq 15 \leq 20$ is true. Therefore,the correct option is $A$.

The maximum number of equivalence relations on the set $A = \{1, 2, 3, 4\}$ is $N$. Then -

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