Question

Determine the number of different equivalence relations on a set with four elements by listing them.

Answer 1

There are 21 different equivalence relations on a set with four elements.

Step-by-step explanation:

An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive. For a set with four elements, we can determine the number of different equivalence relations by listing all possible combinations of the elements.

Let's consider a set with four elements: {a, b, c, d}.

1. First, we can have the empty relation, which is always an equivalence relation.
2. Next, we can have each element related to itself, resulting in four pairs: (a, a), (b, b), (c, c), (d, d).
3. We can also have all possible pairs of the four elements, resulting in six pairs: (a, b), (a, c), (a, d), (b, c), (b, d), (c, d).
4. Finally, we can have both the pairs of each element related to itself and all possible pairs of the four elements, resulting in a total of ten pairs.

Therefore, the number of different equivalence relations on a set with four elements is 1 + 4 + 6 + 10 = 21.