Let S = {1, 2, 3, 4}. How many equivalence relation are there and describe them?

Question

Question

Let $S = \{1, 2, 3, 4\}$.

How many equivalence relation are there and describe them?

I know that 1~2 and 2~3. I also know that there are 2 equivalence relation but how do I prove that and describe them properly.

Answer 1

The equivalence relations correspond to the partitions of the set and are counted by the Bell numbers.