Determining number of different equivalence relations in a set with 4 elements

Question

I have a set with 4 elements.

Let A be $A=\{a,b,c,d\}$

How would I find number of different equivalence relations in this set?

1. Should I use Bell's number theorem in which n would be 4?
2. Should I solve this with the use of combinatorics?

Note however, that this is obviously solvable by logical reasoning, but I need a simple proof or with other words, quick and easy calculation.

Answer 1

Both are the same, really: Bell numbers are defined as counting combinatorial objects. Which one is a "correct" solution depends on your context, i.e., what you are supposed to know/use. As 4 is small, you can even use brute force and list all possible partitions...