First slide

Relations XII

Question

Let X = {1, 2, 3, 4}. The number of equivalence relations that can be defined on X is

Moderate

A

10
B

15
C

16
D

8

Solution

The number of equivalence relations
$B_{k} \sum_{n = 0}^{k - 1} (\begin{matrix} k - 1 \\ n \end{matrix}) B_{n}; B_{0} = 1, B_{1} = 1, B_{2} = 2, B_{3} = 5$
$B_{4} = (\begin{matrix} 3 \\ 0 \end{matrix}) B_{0} + (\begin{matrix} 3 \\ 1 \end{matrix}) B_{1} + (\begin{matrix} 3 \\ 2 \end{matrix}) B_{2} + (\begin{matrix} 3 \\ 3 \end{matrix}) B_{3}$
$= 1 + 3 + 3 \times 2 + 1 \times 5 = 15$

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